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2013
Interaction of ocean waves with oscillating water
column wave energy convertors
Jean-Roch Pierre Nader
University of Wollongong
Recommended Citation
Nader, Jean-Roch Pierre, Interaction of ocean waves with oscillating water column wave energy convertors, Doctor of Philosophy
thesis, School of Mathematics and Applied Sciences, University of Wollongong, 2013. http://ro.uow.edu.au/theses/4183
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School of Mathematics and Applied Statistics
Interaction of Ocean Waves with Oscillating Water
Column Wave Energy Convertors
Jean-Roch Pierre Nader
This thesis is presented as part of the requirements for the
award of the Degree of
Doctor of Philosophy
from
University of Wollongong
February 2013
ABSTRACT
The last two decades have seen increasing interest in renewable energy
technologies in response to the pollution effects from extensive use of fossil fuels
and global warming. It is claimed by some that ocean wave energy alone could
potentially supply the worldwide need for electricity, making it a significant clean
energy resource. However, wave energy is still at an early stage of research and
development and only a few companies are at pre-commercial stage in the
development of their technologies.
With recent increases in computer performance, well-developed mathematical
knowledge in hydrodynamics and advances in the development of numerical
methods, numerical models have become an accurate, low-cost and fast tool in the
research and development of hydrodynamic systems at sea. In the case of ocean
wave energy systems, numerical analysis may be used in the design and optimisation
of such devices.
The focus of the present body of work is on the analysis of the performance
Oscillating Water Column (OWC) devices. An OWC device is essentially a surfacepiercing chamber with a submerged opening in which the free surface moves as a
result of the interaction between the incident wave field and the device. The free
surface movement induces a flow of air through an air turbine connecting the
chamber to the outside ambient air. A generator is then attached to the air turbine
allowing the production of electricity.
Even under linear water wave theory, the wave diffraction and radiation
situation presented by a fixed OWC system has a number of unique features. The
free surface within the OWC represents a complex boundary condition compared to
the remainder of the free surface, which is simply at atmospheric pressure. This
boundary condition includes the influences of a dynamic pressure which is a function
of the flow of air induced by the oscillations of the water free surface, the
compressibility of the air contained in the chamber, and the characteristics of the
turbine system. These turbine characteristics may also be changed in real time so as
to optimise the energy output of the system. Moreover, the maximum power output,
resonant frequency and power bandwidth of the system are all highly dependent on
i
the physical properties of the device. Using a new Finite Element Method (FEM)
model developed by the present author, the optimum turbine parameters have been
determined through the computation of the hydrodynamic properties of the system,
and the effects of the OWC physical attributes have been studied in order to develop
the most efficient design.
Multi-chamber OWC devices or ‘farms’ of OWCs are more likely to be
deployed than a single device in order to harness the maximum available energy in a
region and to facilitate installation and electrical power transmission. Analysis of this
situation involves additional complicating factors since the diffraction, radiation and
interaction between the devices need to be accurately modelled. In the course of the
present study the energy-capture behaviour of OWC devices within an array has been
found to differ significantly to that of a single isolated system. This behaviour is
notably dependent on the spacing between devices relative to the incident
wavelength. Moreover, optimum turbine parameters change depending on the
position of the device in the array and power-capture optimisation of the overall
system needs to take into account each of the device characteristics and all the
interactions between devices.
OWCs may also be configured as floating devices and may be located a
significant distance off-shore to access greater wave power availability. The motion
of the floating structure then becomes an important factor in the power extraction of
the system, where the air flow through the turbines and, therefore, the power
extraction depend on the relative motion between the water and the body. Moreover,
each motion is coupled; higher amplitudes in the water column induce an increase in
the hydrodynamic forces and influence the motions of the body, whereas larger body
motions will increase the generation of radiation waves influencing the motion of the
water inside the chamber. The present study of heaving OWC devices also shows
that mooring properties can have a significance influence on the energetic
characteristic of the system and become an important element in the design of such
systems.
Most of the present WEC studies were performed using linear water wave
theory. The validity of this theory is dependent on the assumption that wave
amplitudes are small in comparison to wavelength. This is not always a valid
ii
assumption in practice. One of the main focuses of the research has been to extend
the model up to the second-order Stokes’ wave theory in order to determine, for the
first time, non-linear effects on the behaviour of fixed and heaving OWC devices. It
is found that second-order terms become especially important around the natural
resonance frequencies of the system. As a consequence, these effects could induce an
important decrease in the mean power output of the device relative to the incident
wave power.
The development of the new 3D FEM model for the analysis of OWC devices
by the present author represents a contribution to our theoretical knowledge and
understanding of these OWC systems in an area that has received scant attention in
the past. While, the results presented in the thesis are focused on relatively simple
OWC geometries, the model has also been applied to assist in the development of
more complex and practical systems for industry.
iii
ACKNOWLEDGEMENT
First, I wish to express my sincere and profound gratitude to my supervisor,
Professor Song-Ping Zhu and my co-supervisor Professor Paul Cooper for letting me
join the exciting ARC project which my PhD study has been part of, at the
University of Wollongong. I also sincerely thank them for all their help, the trust in
my competencies and the support they have given me.
My research was supported by Oceanlinx Ltd, which was the industry partner
in an Australian Research Council (ARC) grant project (LP0776644) entitled “Wave
to Wire – Optimising Hydrodynamic Performance and Capture Efficiency of Next
Generation Ocean Wave Energy Systems” in collaboration with the University of
Wollongong (UOW). I would like to thank Oceanlinx Ltd for their financial support
of this ARC project and particularly thank Dr Tom Denniss and Dr Scott Hunter for
their personal support of my research throughout the project.
I would like to thank my teachers and my previous supervisors for their
inspiration and their priceless knowledge they have offered me.
I also would like to acknowledge all the persons whose smiles, kindness or
compassion have contributed in making my days, months and years brighter.
Finally, I wish to give a big thank you, from the deep of my heart, to all my
family, my grand-parents, my parents, my sister and brother, my partner, my
extended family and all my friends for their precious love and infinite support which
have made it possible, for me, to follow my dreams. I am an amazingly lucky person.
Thank you.
iv
TABLES OF CONTENTS
Abstract ........................................................................................................................ i
Acknowledgement ..................................................................................................... iv
Tables of contents ....................................................................................................... v
List of Figures ............................................................................................................ ix
List of Tables ............................................................................................................ xv
1 Introduction ............................................................................................................. 1
1.1
Renewable Energy ......................................................................................... 1
1.2
ARC Linkage Project ..................................................................................... 1
1.3
The OWC Device ........................................................................................... 2
1.4
Numerical Modelling ..................................................................................... 5
1.5
Thesis Structure ............................................................................................. 9
2 A Finite Element Study of the Efficiency of Arrays of Oscillating Water
Column Wave Energy Converters .......................................................................... 12
2.1
Introduction .................................................................................................. 12
2.2
Formulation .................................................................................................. 12
2.3
The Finite Element Model ........................................................................... 17
2.4
Results and Discussion ................................................................................ 20
2.4.1
Single Isolated OWC Device ............................................................. 20
2.4.2
Arrays of OWC Devices .................................................................... 22
2.4.2.1
Mean Efficiency ......................................................................... 25
2.4.2.2
Capture Width Efficiency .......................................................... 34
2.5
Performance Optimisation ........................................................................... 35
2.6
Conclusion ................................................................................................... 36
3 Hydrodynamic and Energetic Properties of a Single Fixed Oscillating Water
Column Wave Energy Converter ........................................................................... 39
3.1
Introduction .................................................................................................. 39
3.2
Formulation .................................................................................................. 39
3.2.1
General Boundary Conditions ............................................................ 39
3.2.2
Turbine Characteristics and Dynamic Pressure ................................. 42
3.3
Analysis........................................................................................................ 45
v
3.4
The Finite Element Model ........................................................................... 49
3.5
Results and Discussion ................................................................................ 52
3.5.1
Dimensionless Parameters ................................................................. 52
3.5.2
Properties and Air Compressibility .................................................... 53
3.5.3
Physical Properties Effects ................................................................. 62
3.6
3.5.3.1
Draft ........................................................................................... 62
3.5.3.2
Inner radius................................................................................. 63
3.5.3.3
Wall Thickness ........................................................................... 66
Conclusion ................................................................................................... 69
4 Hydrodynamic and Energetic Properties of a Finite Array of Fixed Oscillating
Water Column Wave Energy Converters .............................................................. 70
4.1
Introduction .................................................................................................. 70
4.2
Formulation .................................................................................................. 70
4.2.1
Boundary Conditions ......................................................................... 70
4.2.2
Turbine Characteristics and Dynamic Pressures ................................ 73
4.3
Analysis........................................................................................................ 74
4.4
The Finite Element Model ........................................................................... 78
4.5
Application ................................................................................................... 80
4.5.1
Dimensionless Parameters ................................................................. 80
4.5.2
A Column of Two OWC Devices ...................................................... 81
4.5.2.1
Analysis ...................................................................................... 81
4.5.2.2
Results and Discussions ............................................................. 85
4.5.3
4.5.3.1
Analysis ...................................................................................... 89
4.5.3.2
Results and Discussions ............................................................. 92
4.5.4
4.6
A Row of Two OWC Devices ........................................................... 89
Two Rows and Two Columns of OWC Devices ............................... 95
4.5.4.1
Analysis ...................................................................................... 95
4.5.4.2
Results and Discussions ........................................................... 102
Conclusion ................................................................................................. 102
5 Hydrodynamic and Energetic Properties of a Moored Heaving Oscillating
Water Column Wave Energy Converter ............................................................. 104
5.1
Introduction ................................................................................................ 104
vi
5.2
Formulation ................................................................................................ 105
5.2.1
General Boundary Conditions .......................................................... 105
5.2.2
Heave Motion and Dynamic Pressure .............................................. 108
5.3
Analysis...................................................................................................... 112
5.4
The Finite Element Model ......................................................................... 118
5.5
Results and Discussion .............................................................................. 120
5.5.1
Non-Dimensional Parameters .......................................................... 120
5.5.2
Properties and Air Compressibility .................................................. 121
5.5.3
Effects of the Mooring Restoring Force Coefficient........................ 130
5.6
Conclusion ................................................................................................. 134
6 Power Extraction of a Fixed Oscillating Water Column Device under Weakly
Nonlinear Waves .................................................................................................... 136
6.1
Introduction ................................................................................................ 136
6.2
Formulation ................................................................................................ 137
6.2.1
Stokes’ Wave Expansion.................................................................. 137
6.2.2
Boundary Conditions ....................................................................... 140
6.2.3
Expression for the Pressure .............................................................. 144
6.3
Analysis...................................................................................................... 145
6.4
The Finite Element Model ......................................................................... 149
6.5
Results and Discussion .............................................................................. 153
6.5.1
Dimensionless Parameters ............................................................... 153
6.5.2
Results and Discussions ................................................................... 154
6.6
Conclusion ................................................................................................. 161
7 Power Extraction of a Heaving Oscillating Water Column Device under
Weakly Nonlinear Waves ...................................................................................... 163
7.1
Introduction ................................................................................................ 163
7.1.1
Heaving Motion ............................................................................... 164
7.1.2
Boundary Conditions ....................................................................... 166
7.1.3
Equation of Motion and Turbine Characteristics ............................. 168
7.2
Analysis...................................................................................................... 172
7.3
The Finite Element Model ......................................................................... 177
7.4
Results and Discussion .............................................................................. 178
vii
7.4.1
Dimensionless Parameters ............................................................... 178
7.4.2
Results and Discussions ................................................................... 179
7.5
Conclusion ................................................................................................. 183
8 Conclusions and recommendations ................................................................... 185
References ............................................................................................................... 192
viii
LIST OF FIGURES
Figure 1.1: Drawing from 1920 showing Mr. Bochaux-Praceique’s device. Special
gratitude to Power Magazine, which allowed the reprinting of this drawing
(Palme (1920)). .................................................................................................... 3
Figure 1.2: The PK1 OWC prototype from Oceanlinx Ltd that was located at Port
Kembla, NSW, Australia in 2005. The author gratefully acknowledges
Oceanlinx Ltd for allowing the inclusion of this figure. ...................................... 5
Figure 2.1: Schematic diagram of a single isolated OWC device.............................. 13
Figure 2.2: a) Dimensionless pneumatic damping coefficient
axis) and corresponding
versus
(lower
, (upper axis); b) Comparison of the efficiency
obtained using equation (2.16) and the efficiency obtained using the Evans &
Porter (1997) method. ........................................................................................ 21
Figure 2.3: Schematic diagram of a) an array of 4 OWC devices, and b) an array of 9
OWC devices. .................................................................................................... 23
Figure 2.4: Total wave amplitude around and in the four OWC device array for a
spacing of a)
incident wave is
= 1, b)
= 2, c)
= 5 and d)
= 0 and the frequency is
plotted at intervals such that
= 10. The angle of the
= 3.4. The contour lines are
=0.1. ........................................................ 27
Figure 2.5: Comparison of the total mean efficiency
for array spacing of
=
1, 2, 5, 10 with the efficiency of the Single Isolated OWC Device (SIOD) versus
and corresponding
for
= 0. a) corresponds to the four devices
arrangements and b) corresponds to the results obtained for the nine devices
arrangements. ..................................................................................................... 28
Figure 2.6: Comparison of the total mean efficiency
for array spacing of
=
1, 2, 5, 10 with the efficiency of the Single Isolated OWC Device (SIOD) versus
and corresponding
for
= π/8. a) corresponds to the four devices
arrangements and b) corresponds to the results obtained for the nine devices
arrangements. ..................................................................................................... 29
Figure 2.7: Comparison of the total mean efficiency
for array spacing of
=
1, 2, 5, 10 with the efficiency of the Single Isolated OWC Device (SIOD) versus
and corresponding
for
= π/4. a) corresponds to the four devices
ix
arrangements and b) corresponds to the results obtained for the nine devices
arrangements. ..................................................................................................... 30
Figure 2.8: Comparison of the capture width efficiency
for array spacing of
= 1, 2, 5, 10 with the efficiency of the Single Isolated OWC Device (SIOD)
versus
and corresponding
for
= 0. a) corresponds to the four devices
arrangements and b) corresponds to the results obtained for the nine devices
arrangements. ..................................................................................................... 31
Figure 2.9: Comparison of the capture width efficiency
for array spacing of
= 1, 2, 5, 10 with the efficiency of the Single Isolated OWC Device (SIOD)
versus
and corresponding
= π/8. a) corresponds to the four devices
for
arrangements and b) corresponds to the results obtained for the nine devices
arrangements. ..................................................................................................... 32
Figure 2.10: Comparison of the capture width efficiency
for array spacing of
= 1, 2, 5, 10 with the efficiency of the Single Isolated OWC Device (SIOD)
versus
and corresponding
= π/4. a) corresponds to the four devices
for
arrangements and b) corresponds to the results obtained for the nine devices
arrangements. ..................................................................................................... 33
Figure 2.11: Comparison of the efficiency
obtained for the OWC device a)
number 1 and b) number 2 of the four device array with pneumatic damping
coefficients of
= 1, 0.5, 0.7, 0.9, 1.1, 1.3 and 1.5 versus
corresponding
spacing is
. The angle of the incident wave is
and
= 0 and the array
= 5. ............................................................................................. 37
Figure 3.1: Schematic diagram of a single isolated OWC device.............................. 40
Figure 3.2: Example of a mesh used around the OWC device .................................. 49
Figure 3.3: Wave amplitudes around and in the OWC device from a) the diffraction
problem
frequency is
are
, b) the radiation problem
and c) the overall problem . The
= 3, and the parameters of the turbine and air compressibility
and
................................................................................. 53
Figure 3.4: Dimensionless radiation conductance
air compressibility parameter
0, 1, 3, 6.82 and 15
frequency
, radiation susceptance
and
for different chamber volume of air,
The
-axis represents the dimensionless
. ..................................................................................................... 55
x
Figure 3.5: Non-dimensional amplitudes, versus
flux
and the total volume flux
using the turbine parameter
different chamber volumes of air,
parameters
and
, of the diffracted wave volume
0, 1, 3, 6.82, 15
volumes of air
0, 1, 3, 6.82, 15
capture width
for different
for different chamber
and dimensionless maximum
. .......................................................... 58
Figure 3.7: Dimensionless a) radiation conductances
versus
0.3 and 0.5.
and using the
. ................................................................... 57
Figure 3.6: Dimensionless optimum capture width
susceptances
for
and b) radiation
for different OWC device draft
0.2 and
0.1, 0.15, 0.2,
0.25. ............................................................ 60
Figure 3.8: Dimensionless a) amplitudes of the complex excitation coefficient
and b) maximum capture widths
draft
versus
0.1, 0.15, 0.2, 0.3 and 0.5.
for different OWC device
0.2 and
0.25. ................ 61
Figure 3.9: Dimensionless a) radiation conductances
and b) radiation
susceptances
versus
0.15, 0.2, 0.25 and 0.3.
for different OWC device inner radius
0.2 and
0.1,
0.05. .............................. 64
Figure 3.10: Dimensionless a) amplitudes of the complex excitation coefficient
and b) maximum capture widths
inner radius
versus
for different OWC device
0.1, 0.15, 0.2, 0.25 and 0.3.
0.2 and
0.05. .................................................................................................................... 65
Figure 3.11: Dimensionless a) radiation conductances
susceptances
0.25, 0.3 and 0.4.
versus
and b) radiation
for different OWC device outer radius
0.2 and
0.21,
0.2. ...................................................... 67
Figure 3.12: Dimensionless a) amplitudes of the complex excitation coefficient
and b) maximum capture widths
outer radius
0.21, 0.25, 0.3 and 0.4.
versus
for different OWC device
0.2 and
0.2. .............. 68
Figure 4.1: Examples of the meshes used around the OWC devices for the different
problems studied in Section 4.5. a) A column of two OWC devices, b) A row of
two OWC devices and c) Two rows and two columns of OWC devices. These
meshes are related to the non-dimensional frequency
. ......................... 78
Figure 4.2: Schematic Diagram of the arrangement of a column of two identical,
fixed, cylindrical OWC devices. ........................................................................ 81
xi
Figure 4.3: Dimensionless free surface amplitudes attached to a) the radiation
problem 1,
problem
, b) the sum of the radiation problems,
, c) the diffraction
, and d) the overall problem, . The frequency is
parameters of the turbine and air compressibility are
Figure 4.4: Radiation conductances
and
and
and
of the SIOD, versus
. . 84
and radiation susceptances
compared with the radiation conductance
susceptance
= 3, and the
and radiation
. ........................................................... 86
Figure 4.5: Amplitude of the dimensionless complex excitation coefficients
compared with the one from the SIOD, and amplitude of dimensionless volume
flux
for different turbine and compressibility parameters,
and
versus
. ...................................................................................................................... 87
Figure 4.6: Dimensionless optimum capture width
for different air chamber
volume of air and maximum dimensionless capture
the
compared with
of the SIOD and the capture width
parameters
and
versus
obtained with the
. .............................................................. 88
Figure 4.7: Schematic Diagram of the arrangement of a row of two identical, fixed,
cylindrical OWC devices. .................................................................................. 89
Figure 4.8: Dimensionless free surface amplitudes attached to a) the radiation
problem 1,
problem,
, b) the sum of the radiation problems,
, c) the diffraction
, and d) the overall problem, . The frequency is
parameters of the turbines and air compressibility are
and
= 3, and the
,
,
. ............................................................................. 93
Figure 4.9: a) Dimensionless amplitude of the complex excitation coefficients
,
compared with
of the SIOD. b) Dimensionless parameters of the
,
and dimensionless parameters of air
turbine
compressibility
dimensionless frequency
and
frequency
-axes represents the
............................................................................... 94
Figure 4.10: Dimensionless capture widths
compared with
. The
,
,
of the SIOD. The -axes represents the dimensionless
. ..................................................................................................... 95
Figure 4.11: Schematic Diagram of the two-rows and two-columns arrangement of
four identical, fixed, cylindrical OWC devices. ................................................. 95
xii
Figure 4.12: Dimensionless free surface amplitudes attached to a) the radiation
problem 1,
problem
, b) the sum of the radiation problems,
, c) the diffraction
, and d) the overall problem, . The frequency is
parameters of the turbines and air compressibility are
and
= 3, and the
,
,
. ............................................................................. 98
Figure 4.13: a) Dimensionless amplitude of the complex excitation coefficients
,
and
related to the SIOD. b) Dimensionless radiation
conductances and radiation susceptances
,
and
SIOD. The -axes represents the dimensionless frequency
Figure 4.14: a)
,
. b)
dimensionless frequency
Figure
4.15:
a)
,
,
for the
. ........................ 99
. The
-axes represents the
............................................................................. 100
Dimensionless
parameters
of
the
turbine
,
and dimensionless parameters of air compressibility
and
.
b)
,
Dimensionless
,
capture
and
widths
,
related to the SIOD. The -
axes represents the dimensionless frequency
. ............................................ 101
Figure 5.1: Schematic diagram of a single floating isolated OWC device .............. 105
Figure 5.2: Example of a mesh used around the OWC device ................................ 118
Figure 5.3: Dimensionless free surface amplitudes a)
The frequency is
, b)
, c)
and d) .
= 3, and the parameters of the turbine, air compressibility
and the mooring restoring force coefficient are
,
and
0N.m-1. ............................................................................................................. 122
Figure 5.4: Real and imaginary parts of the dimensionless hydrodynamic coefficients
a)
and b)
, against
. .................................................................... 124
Figure 5.5: Real and imaginary parts of the dimensionless hydrodynamic coefficients
a)
and b)
against
. ..................................................................... 125
Figure 5.6: Real and imaginary parts of the dimensionless hydrodynamic coefficients
a)
and b)
and
against
. ............................................................. 126
Figure 5.7: a) Non-dimensional volume flux amplitudes
b) Dimensionless amplitude of the total volume flux
0, using the turbine parameter
,
and
when
0.
for different cases, for
for different chamber volumes of
xiii
air,
0, 5, 10, 50
and using the parameters
The -axis represents the dimensionless frequency
and
and
Figure 5.8: Dimensionless optimum capture width
.
0. ............. 129
for different chamber
volumes of air
0, 5, 10, 50
capture width
. The -axis represents the dimensionless frequency
and
and dimensionless overall maximum
0. ..................................................................................................... 130
Figure 5.9: a) Dimensionless overall radiation conductance
overall radiation susceptances
0, 0.1, 0.5, 1, 5
and b) dimensionless
for different restoring force coefficients
compared with the fixed device results. The -axis
represents the dimensionless frequency
..................................................... 132
Figure 5.10: a) Dimensionless total volume flux amplitudes
0 and for
width
and
for
coefficients
for
and
and b) dimensionless optimum capture
0 and
0, 0.1, 0.5, 1, 5
for different restoring force
compared with the fixed device results.
The -axis represents the dimensionless frequency
. .................................. 133
Figure 6.1: Schematic diagram of the OWC device ................................................ 138
Figure 6.2: Comparison of the free surface amplitudes
,
and
resulting
from the model with the results from Chau & Eatock Taylor (1992), around the
circumference of a cylinder.
,
m,
and
.
is the
azimuth of the cylindrical coordinates in degrees. ........................................... 151
Figure 6.3: Amplitude of the dimensionless volume fluxes
different pneumatic damping coefficients
versus
and
0, (
0) and
. .......................................................................................... 155
Figure 6.4: Amplitude of the dimensionless volume fluxes versus
, and
for
for
0 and b)
,
,
. a)
, and
,
for
. ........................................................................................................... 156
Figure 6.5: Dimensionless free surface amplitudes around and inside the OWC
device. a)
(
, b)
0) and c)
considering the pneumatic damping coefficient
, d)
using the pneumatic damping coefficient
0,
.
3. ............................................................................................................. 159
xiv
Figure 6.5: Dimensionless mean hydrodynamic power extracted
various incident wave amplitudes
and
for
0.01, 0.02, 0.05 and 0.1 versus
The pneumatic damping coefficient is
.
. ..................................... 160
Figure 7.1: Schematic diagram of the OWC device. ............................................... 164
Figure 7.2: Amplitude of the dimensionless volume fluxes versus
for different pneumatic damping coefficients
; b)
and
for
,
, and
for
. a)
and
0, (
0; b)
0) and
,
,
,
. ............................................................................ 180
Figure 7.3: Dimensionless free surface amplitudes around and inside the OWC. a)
, b)
and c)
considering the pneumatic damping coefficient
, d)
using the pneumatic damping coefficient
.
Figure 7.4: Dimensionless mean hydrodynamic power extracted
various incident wave amplitude
0N.m-2)
0, (
3. . 181
and
for
0.01, 0.02, 0.05 and 0.1 versus
. 182
LIST OF TABLES
Table 2.1: Influence of the radiation boundary parameter
on the efficiency
obtained for the OWC device number 1 of the four device array configuration.
The angle of the incident wave is
= π/4 and the array spacing is
xv
= 10. .. 24
1
1 INTRODUCTION
1.1
Renewable Energy
Over the past two decades, nations worldwide have been looking for new
energy sources in order to slow down the effect of global warming induced by the
extensive use of fossil fuels. A large number of renewable energy sources are
currently being researched, developed and applied. There are six main areas of
renewable energy technology: bioenergy, direct solar energy, geothermal energy,
hydropower, ocean energy and wind energy. An overview of these technologies can
be found in the Special Report on Renewable Energy Sources and Climate Change
Mitigation (SRREN), IPCC (2011), agreed and released by the Intergovernmental
Panel on Climate Change (IPCC).
Ocean energy can be separated into wave energy, tidal range, tidal currents,
ocean currents, ocean thermal energy conversion and salinity gradient. The overall
potential of ocean energy largely exceeds the present energy requirements. Wave
energy alone has been estimated at around twice the overall electricity supply in
2008, (cf. Lewis et al. (2011)), making it a significant clean source of energy. Unlike
wind energy technologies, wave energy is still in the early stage of research and
development. More than 50 different systems have been experimented but only a
handful of these have been tested in full scale. Reviews of these technologies can be
found in Clément et al. (2002), Cruz (2008), Falnes (2007), Khan & Bhuyan (2009)
and Falcão (2010). However, with the help of government initiatives the technology
is developing quickly around the world. As an example, the Australian
Commonwealth Scientific and Industrial Research Organisation (CSIRO) released an
analysis, CSIRO (2012), stating that wave energy could contribute up to 11 per cent
of the total Australian electrical power by 2050. Recently three Australian wave
energy
companies
(Carnegie Wave Energy Ltd,
Oceanlinx Ltd
and
BioPower Systems Pty Ltd) were granted funds from the Australian Renewable
Energy Agency (ERP (2012)) to support pilot projects around Australia.
1.2
ARC Linkage Project
The current thesis was completed by PhD student, Jean-Roch Nader. The PhD
position has taken place under an Australian Research Council (ARC) grant
2
(LP0776644) between the University Of Wollongong (UOW) in conjunction with
industry partner Oceanlinx Ltd.
Oceanlinx Ltd (originally known as Energetech) at the time of writing was one
of the most prominent Australian and international companies in the field of wave
energy. Over the previous 15 years, Oceanlinx Ltd developed, deployed and operated
three prototypes of their technology in the open ocean. Their technology is based on
the Oscillating Water Column (OWC) concept. An OWC device is essentially a
surface-piercing chamber with a submerged opening in which the free surface moves
as a result of the interaction between the incident wave field and the device. The free
surface movement induces a flow of air through an air turbine connecting the
chamber to the outside ambient air. A generator is then attached to the air turbine for
the production of electricity.
The ARC Linkage project was entitled “Wave to Wire – Optimising
Hydrodynamic Performance and Capture Efficiency of Next Generation Ocean Wave
Energy Systems”. The aim of the project was to improve the performance of ocean
wave energy technologies for electricity generation and desalination. The main focus
of the project was the application of new theoretical and experimental approaches to
the development of next-generation near-shore OWC Wave Energy Convertors
(WECs). The research performed in this project was focused on assisting the partner
organisation, Oceanlinx Ltd, in developing one of the most effective and
economically attractive wave energy conversion systems in the world.
The role of the PhD student, within this project, was to develop a new three
dimensional model based on the Finite Element Method (FEM) in order to study the
complex hydrodynamic and energetic performance of OWC devices in waves.
Publications related to the overall project includes Stappenbelt & Cooper
(2009), (2010a), (2010b), Stappenbelt et al. (2011), Luo et al. (2012), Nader et al.
(2011), (2012b), Nader et al. (2012a), (2014) and Nader (2014).
1.3
The OWC Device
The Oscillating Water Chamber concept was, in fact, one of the first practical
applications of wave energy conversion to be carried out. As documented by Palme
(1920), in 1910 Mr Bochaux-Praceique attached an air turbine to a vertical bore hole
connected to the sea in Royan, near Bordeaux in France, and it is claimed that he
3
could supply his house with 1kW of electricity. A representation of the device can be
seen in Figure 1.1.
Figure 1.1: Drawing from 1920 showing Mr. Bochaux-Praceique’s device. Special
gratitude to Power Magazine, which allowed the reprinting of this drawing (Palme
(1920)).
However, after the First World War, petroleum became the main source of
energy and the development of other resources faded away. The most extensive
research into wave energy since the Second World War was conducted by a former
Japanese naval commander, Yoshio Masuda, who did his first sea tests in 1947. He is
regarded as the pioneer of modern wave-energy development. He tested a large
number of systems including the first floating OWC device. His work was mostly
focused on the powering of navigation buoys and hundreds of these self-powered
buoys were sold in Japan and America (Masuda (1971)).
In Europe, only the 1973 oil crisis induced a renewed interest in wave energy
systems. Governments started allocating funds for research in this area and extensive
programs of research were conducted (cf. reviews from Shaw (1982) and Lewis
(1985)). After the 1980s, the oil price decreased and so did wave-energy funds. The
4
technology was even considered unviable by some governments. It is only since
1991, when the European Commission included wave energy in their research and
development program on renewable energies, that further projects could be carried
out (see review by Clément et al. (2002)). More recently, Canada, the USA and
Australia have also started to get involved in the field of wave energy (see Hayward
& Osman (2011), Previsic et al. (2009), CSIRO (2012)).
One key part in the development of wave-energy research, since the 1973 oil
crisis,
was
the
introduction
of
theoretical
hydrodynamics
by renowned
mathematicians. The first theory related to a fixed OWC device was developed by
Evans (1978), under linear water wave theory. He extended its “theory for wavepower absorption by oscillating bodies”, in Evans (1976), by considering the free
surface inside the OWC chamber as a weightless piston. He then improved his theory
by allowing the free surface to oscillate under the application of an uniform pressure
in Evans (1982). Sarmento & Falcao (1985), investigated the effect of air
compressibility inside the OWC chamber while Falnes & McIver (1985) looked at
the “surface wave interactions with systems of oscillating bodies and pressure
distributions” introducing a theory for floating OWC devices as well as for
interactions between devices. There are now a large numbers of studies based on
these theories.
A difficulty in the application of OWC devices has been the design of an
efficient air turbine. Unlike wind turbines, the air flow passing through the OWC
turbine reverses with time. Masuda (1979) was involved with the construction of a
large barge, named Kaimei, used to test several types of turbine but could not obtain
a satisfactory power output level. Most of the air turbines currently used are based on
the Wells (1976) turbine. The effectiveness of this turbine comes from its ability to
continuously rotate in one direction independently of the direction of the air flow.
Improvements and modifications have been investigated by several researchers and
an in-depth review of this research on turbines can be found in Curran & Folley
(2008).
The OWC is certainly one of the wave energy systems that has seen the highest
number of applications. The device can be shore-based as the Pico plant in Portugal
(Falcão (2000)), integrated in a breakwater as in the harbour of Sakata, Japan
5
(Takahashi et al. (1992)), near-shore bottom-standing like the PK1 prototype (Figure
1.2) that was tested off the coast of Port-Kembla in 2005 by Oceanlinx Ltd, or
floating like the Mighty Whale developed by the Japan Marine Science and
Technology Center (Washio et al. (2000)), to name but a few. Further reviews of the
different devices tested can be found in Falnes (2007).
Figure 1.2: The PK1 OWC prototype from Oceanlinx Ltd that was located at Port
Kembla, NSW, Australia in 2005. The author gratefully acknowledges Oceanlinx Ltd
for allowing the inclusion of this figure.
1.4
Numerical Modelling
With the constant increase in computer performance, well-developed
mathematical knowledge in hydrodynamics and advancing numerical methods,
numerical models have become an accurate, low-cost and fast tools for research and
development of systems at sea. In the case of wave-energy systems, numerical
models can be a benchmark in the testing, designing and optimisation of such
devices.
The overall aim of the work described in this thesis was to develop a new
numerical model in order to study the hydrodynamic and energetic behaviour of
OWC devices at sea.
6
Several methods have been developed for the study of OWC devices using
linear water wave theory. One of the simplest methods is based on mechanical
models where the OWC device is described as a system of lumped-masses. Using
such a method, Folley & Whittaker (2005) looked at the effect of plenum chamber
volume and air turbine hysteresis on the optimal performance of oscillating water
columns, while Stappenbelt & Cooper (2010b) investigated such optimisation for a
floating device. Such simplified approaches provide indications of device
performance trends and are useful in the preliminary design and model testing and
development phases. However, these types of model do not analyse the full
hydrodynamic complexity of the situation and the hydrodynamic coefficients
required as inputs need to be approximated by other means.
A more advanced method has been the development of analytical solutions
satisfying the boundary value problems imposed by OWC device systems. A large
amount of work in this field can be attributed to Mavrakos (1985), (1988), (2000),
(2005) which includes wave loads, diffractions, hydrodynamic coefficients and
interactions between stationary, freely floating or heaving cylindrical bottomless
cylinders with finite wall thickness. However, no pressure effects were considered.
Other analytical solutions were developed by Zhu & Mitchell (2009) who reexamined a classic case presented by Garrett (1970) for the case of diffraction of
plane waves by a fixed OWC. They later extended their study to the case of
radiational waves created by the pressure exerted inside the device chamber (Zhu &
Mitchell (2011)). Martins-Rivas & Mei (2009a), (2009b) applied the well-known
Eigen function expansion method to a single circular fixed OWC at the tip of a thin
breakwater and along a straight coast, aiming to determine the necessary conditions
to achieve maximum power take-off. Such methods can be relatively accurate as
numerical errors only appear through the necessary truncation of the analytical series
and they are also quite efficient in terms of computing time. However, these methods
are mostly limited to simple device geometries and flat bathymetry due to the
complexity of finding an analytical solution to the hydrodynamic problem.
In order to overcome such difficulties, more complex hydrodynamic numerical
models have been employed. These models are commonly based on the Finite
Element Method (FEM) (cf. Garrigues (2002) and Zienkiewicz et al. (2005)) and the
7
Boundary Element Method (BEM) (cf. Newman (1992)) or a combination of these.
The commercial 3D BEM model WAMIT has often been employed to model wave
power extraction in the past. Sykes et al. (2007), (2009) studied fixed and freely
floating OWC devices. Delauré & Lewis (2003) also developed a model of a fixed
OWC device including dynamic chamber pressure.
Non-linear numerical models are also being developed to investigate waveOWC device interaction. Weber & Thomas (2001a), (2001b) included a 2D linear
hydrodynamic and non-linear aerodynamic coupling in an OWC device in order to
better simulate the effect of the turbine. 2D fully non-linear Euler models have been
applied to an OWC device by Luo et al. (2012), to study the effect of increasing
wave amplitudes and by Mingham et al. (2003) in a specific study of the LIMPET
OWC device. Finally, 3D models based on the incompressible Navier-Stokes
equations have not yet been applied to OWC case but have the potential to be used in
this domain for specific studies. These models such as the ones developed by Causon
et al. (2008) and Agamloh et al. (2008), who studied floating type WECs, have the
potential to include loads under extreme wave conditions and the effect of
turbulence. It was found experimentally by Fleming et al. (2013) using a PIV
measurement system that turbulence can become especially important around the
entrance of an OWC device of the type used by Oceanlinx. Turbulence can comprise
up to 5% of the total energy flux and a Navier-Stokes based model could have the
potential of studying these effects. However, these non-linear models are usually
time-stepping and require a very large amount of memory and CPU time limiting the
amount of studies and applications they can perform.
The numerical methods used to solve the problem under fully non-linear
theories are often based on the finite differences or finite volumes. The stiffness
matrix resulting from finite differences ore finite volumes are usually larger and
more populated than FEM or BEM which means that they require a larger amount of
memory and CPU time. But under fully non-linear theories, re-meshing at each time
step becomes necessary when using FEM and BEM whereas the meshing in the other
methods can be unchanged by also modelling a section of the air above the water and
using specific techniques to capture the free surface. It follows that, under these
conditions, finite differences ore finite volumes can become more time efficient.
8
Under linear water wave theories, BEM models also have their limitations.
WAMIT has been developed with a source distribution approach where, by use of a
potential flow theory and a boundary integral equation, the strength of the source can
be determined on a set of boundary nodes placed on the boundary of the
computational domain of the problem. As documented in Delauré & Lewis (2003),
difficulties arise when modelling the presence of enclosed domains such as the
chamber of an OWC device where the discontinuity between internal and external
sources can induce non-negligible numerical errors around the resonant frequency.
To account for internal air pressure effects, the problem is divided into the scattering
and radiation problems. When using WAMIT for the radiation problem, the OWC
chamber surface has to be modelled as a weightless deformable piston where modal
shapes have to be specified. Even under linear water wave theory, one might be
concerned as to the ability of the BEM to efficiently deal with extensions to secondorder water wave theory, one of the main foci of the project. Other types of BEM
model use fundamental solutions of the problem. However, this introduces other
types of difficulty. For example, when variable water depth needs to be taken into
consideration, the presence of some volume integrals, as a result of no fundamental
solution being available, compromises the elegance of the method (e.g. Zhu (1993),
Zhu et al. (2000)).
On the other hand, a FEM is based on the discretisation of the entire
computational domain into a finite number of elements where the quantity of interest
is approximated. By use of Green’s theorem, which provides a relation between the
volume integral and the surface boundary integrals enclosing the volume, the
approximation functions can be obtained. The quantity of interest can then be found
directly at any grid point inside the computational domain. The surface integral
includes the various boundary values and seems to be a more direct and less
problematic method to deal with the dynamic pressure inside an OWC chamber and
the implementation of second-order terms. Although a FEM is limited in terms of
computational domain size due to the computing resources required for the
discretisation of the entire volume, it was deemed better suited to the objectives of
the present project as the aim was for model to be general enough to take into
9
consideration variable water depth, coupling with the turbine system and
investigation of weakly non-linear effects.
A number of commercial hydrodynamic FEM models exist such as ANSYS®.
However, when using such software, studies are limited by the content of the
product. The development of a new model allowed a more specific focus on a
particular system and the advancement of uncovered areas of research. In the present
case, this area of research has been the study of the hydrodynamic and energetic
behaviour of OWC devices at sea.
1.5
Thesis Structure
The following six chapters present the application of the newly developed
hydrodynamic FEM model to specific research studies related to OWC devices.
Chapter 2 presents one of the first research studies performed with an earlier
version of the newly developed FEM model. It focuses on the effect of wave
interactions on power-capture efficiency of finite arrays of OWC devices. The model
was applied to a single fixed OWC device with dynamic chamber pressure and
several array configurations of these devices. The resulting power capture efficiency
for a single OWC over a range of wave frequencies is compared to the predictions of
the model proposed by Evans & Porter (1997). Two arrangements of arrays of
cylindrical shaped OWC devices are also examined with the finite element method
developed. Results for various array spacings and directions of the incident wave are
reported and compared with those of a single isolated OWC device.
The study was, however, limited in the description of the hydrodynamic
properties of the system. Moreover, following Evans & Porter (1997), the pneumatic
damping coefficient, reflecting the relationship between the volume flux and the
pressure inside the chamber, was considered positive and real. Such an assumption
overlooks the effect of air compressibility inside the OWC chamber. As shown by
Sarmento & Falcão (1985) and, subsequently, by Martins-Rivas & Mei (2009a),
(2009b), air compressibility can have a non-negligible effect on the power extraction
of an OWC device.
Chapter 3 considers a single fixed OWC device. Air compressibility inside the
chamber is taken into consideration and a more in-depth analysis of the
hydrodynamic characteristics is presented. It is shown that only a few specific
10
hydrodynamic coefficients are needed in order to determine the dynamic and
energetic behaviour of an OWC device in waves. Using this method, it is also
possible to directly derive the closed form of the optimum damping coefficient so as
to obtain maximum hydrodynamic power extraction from the system. This method
also provided the basis for the analysis of more complex systems as in the following
chapters. Finally, the method is applied to a cylindrical fixed OWC device with finite
wall thickness using a newer more efficient FEM model. Special attention is given to
the effect of air-compressibility and physical properties (draft, radius and wall
thickness) in the maximum hydrodynamic power available to the system.
Chapter 4 is the natural extension of the preceding two chapters where the
method developed in Chapter 3 is extended to finite arrays of OWC devices.
Following the interaction theory between oscillating systems introduced by Falnes &
McIver (1985), the different influences between devices are taken into account. Air
compressibility is also considered and a new method for turbine parameter
optimisation is developed. Results from the 3D FEM model for three different
arrangements of cylindrical OWC devices are then presented and discussed.
OWC devices can also be floating structures. The motion of the structure
becomes an important factor in the power extraction of the system. Mooring
properties and air pressure inside the chamber can especially influence these
motions. Chapter 5 looks at a single OWC device that is allowed to heave, i.e. the
motion has one degree of freedom. Direct coupling between the motion of the device,
the pressure inside the chamber, the volume fluxes and the forces is considered. The
3D FEM model is then applied to a heaving cylindrical OWC device with finite wall
thickness in order to the study its dynamic and energetic behaviour. Special focus is
given to the effect of air compressibility, the optimisation of the turbine parameter
and the effects of the mooring restoring force coefficient.
In the previous chapters, different problems related to the behaviour of OWC
devices in waves were examined using linear water wave theory. The validity of
linear theory is associated with the assumption that the wave amplitudes are small in
comparison to their wavelength. Studies based on linear water wave theory are
therefore limited. This is especially the case when an OWC device is placed near
shore where the water depth is shallow enough that some weakly nonlinear effects
11
must be taken into consideration. Moreover, OWC devices are resonant systems and
even for relatively small incident wave amplitudes, significant amplification can be
found in the OWC chamber. In Chapter 6 and Chapter 7, the FEM is extended using
Stokes’ wave theory (Stokes (2009)) up to second order and the second-order 3D
FEM was applied to fixed and heaving cylindrical OWC devices. Second-order
corrections in terms of the overall volume flux inside the chamber and mean power
output are presented and compared with the results obtained from linear water wave
theory.
Chapter 8 summarises the different results presented in the thesis and discusses
the research and development recommended for future studies of OWC devices.
The overall body of work in this thesis covers an intensive and coherent
theoretical study of OWC devices in waves. However, each of the main chapters was
written in a journal article style contrary to a more traditional monograph style. This
format complies with the University Of Wollongong general course rules. This
layout allows each study to be considered independently if desired. On the other
hand, the reader should be aware that under this arrangement a degree of repetition
may appear especially in the description of the system, the general formulation and
the description of the numerical model used to solve the problem.
12
2 A FINITE ELEMENT STUDY OF THE EFFICIENCY OF ARRAYS OF
OSCILLATING WATER COLUMN WAVE ENERGY CONVERTERS
2.1
Introduction
Multi-chamber OWC devices or farms of OWCs are more likely to be
deployed than single devices in order to harness maximum available energy in a
region and to facilitate installation and electrical power transmission. As such, the
performance and behaviour of arrays of devices is also of significant interest. Such
multiple device arrangements can produce strong scattering interactions affecting the
efficiency of each device. Research in this field has mostly been in the context of
offshore platforms and only a few studies directly related to wave energy conversion
exist. Mavrakos & McIver (1997), for example, compared the multiple scattering
method with the plane-wave method to compute the wave forces, hydrodynamic
coefficients and q-factors of a finite array of devices. More recently, Garnaud & Mei
(2010) used an asymptotic theory to study the interactions and their effects on the
power take-off efficiency of a single periodic array of small buoys in a channel. In
term of numerical modelling, Vicente et al. (2009) used the 3D BEM model WAMIT
to investigate a triangular array configurations of point absorbers with a particular
focus on the mooring system behaviour.
In the present chapter, one the first research study performed with an early
version of the FEM model is presented. The study focuses on the effect of wave
interactions on the power capture efficiency for finite arrays of OWC devices. The
results of the model applied to a single fixed OWC device with dynamic chamber
pressure and several array configurations of these devices are presented. The
resulting power capture efficiency for a single OWC over a range of wave
frequencies is compared with the predictions from the model proposed by Evans &
Porter (1997). Two arrangements of arrays of cylindrical shaped OWC devices are
also examined with the finite element method developed. Results for various array
spacings and directions of the incident wave are reported and compared with those of
a single isolated OWC device.
2.2
Formulation
13
Figure 2.1: Schematic diagram of a single isolated OWC device
The OWC device considered in the present study is a truncated hollow cylinder
with a finite wall thickness as illustrated in Figure 1.1. The cylinder is surface
piercing and operates in constant water depth . The inner radius is
radius is . The draft of the cylinder is
and the outer
. A Cartesian coordinate system
with its corresponding cylindrical coordinates
are situated with the origin
coincident with the axis of the cylinder at the mean sea water level and the
-
direction pointing vertically upwards. A monochromatic plane wave of amplitude
and frequency
propagates from
.
Linear water-wave theory is assumed and with the assumptions of irrotational
and inviscid flow, a velocity potential
exists that satisfies the Laplace
equation
Under these assumptions
value
can be expressed using its corresponding complex
as
The computational domain is separated into two regions; an outer region
between a radius of
and
with velocity potential
and an inner region within
14
the radius
with a velocity potential
. The velocity potential
decomposed into the sum of the incident wave velocity potential
potential
can be
and the velocity
induced by the scattering of the wave by the device as
with
In this expression,
equal to
is the gravitational constant and , the wave number, is
. The parameter
is the wavelength. The wave number, , satisfies the
dispersion relation
The general boundary conditions for this problem can be expressed as follows:

In the outer domain
On the sea floor at
=
On the surface, at
=0
≥ ≥
:
The Sommerfeld radiation condition on
when
→ ∞
15

Between the two regions at

In the inner region
≤
=
:
:
At any point on the device walls
where
is the derivative in the direction of the unit vector
normal to the
surface of the wall and pointing outward of the fluid.
On the surface, at
= 0 and ≥ ,
An oscillating pressure,
, is applied in the chamber. The frequency of the
oscillations is considered equal to the incident wave frequency and therefore the
complex pressure,
, can be introduced in the equation
The boundary condition at the free surface inside the chamber, at
= 0 and
≤
, can then be expressed as
where
is the density of the water. The complex pressure,
uniform on the surface
, is also assumed to be
of the water inside the chamber and linearly dependent on
16
the total volume flux inside the chamber. This assumption is justifiable when, for
example, a Wells type turbine is employed for power take-off. For a fixed OWC
device, the volume flux is equal to the volume flux
free surface elevation
induced by the variation of the
inside the chamber.
This assumption then gives
The parameter
is the pneumatic damping coefficient which defines the
pressure in the OWC as a result of the volume flow rate through the air turbine. In
practice,
is determined by the choice of air turbine design and may be controlled
in real time, for example by varying the angular velocity of the turbine rotor (e.g.
Cruz, 2008) and/or the pitch angle of the turbine blades in order to maximise the
energy output of the OWC system (e.g. Gato et al. (1991)). Following Evans &
Porter (1997), we consider the damping coefficient
to be real and positive,
meaning that the turbine does not exhibit any time lag between the volume flux and
the pressure.
The boundary condition (2.13) inside the chamber at
The mean power output
where
= 0 and
≤
becomes
of the device can then be calculated through
is the complex conjugate of
.
We define the capture efficiency, also called the relative capture width (e.g.
Cruz, 2008) as follows
17
where
is the power capture width and
is the mean wave power (averaged over
the wave period) per crest width of a monochromatic plane wave of amplitude
frequency
where
and
propagating in the -direction.
is the group velocity. It should be remarked that due to the diffraction of
the wave around the device, the effective mean power available to the system can be
higher than the mean wave power of the free wave passing through a width of 2 .
Therefore,
2.3
can potentially exceed unity.
The Finite Element Model
A new Finite Element Method (FEM) model has been developed in order to
numerically solve the system described previously. The method is based on the
discretisation of the domain into a finite number of elements. In each element, the
quantity of interest, the velocity potential in this case, is approximated by the sum of
the product of the velocity potential at a finite number of nodes and a shape function
as
The parameter
is the velocity potential inside the element ,
of the velocity potential at the node
element , and
of coordinates
is the shape function.
The equations solved by the model are
is the value
included in the
18
Using Green’s theorem, for any test function
the following
relationship can be derived
where
and
respectively,
are the total volumes of the inner domain and outer domain
and
are the surfaces enclosing the volumes and
unit normal vectors with respect to the surfaces
and
and
are the
, directed to the outside of
the domains. The test functions are chosen to be equal to the total number of shape
functions following the Galerkin method.
Applying Green’s theorem to the two domains using the different shape
functions and incorporating the various boundary conditions leads to a finite number
of linear equations. The number of these equations is equal to the sum of the total
number of nodes of each domain,
where
is an
×
, which can be regrouped in a matrix format as
matrix, called the stiffness matrix,
of potential values at the nodes and
is a
is the
× 1 vector
× 1 vector.
Values of the velocity potential at each node
are therefore obtained directly
by solving this system of equations. The potential and its derivatives can then be
obtained at any position in the domains by the use of the approximated solution
in
each element.
As the computational domain is finite, the following radiation boundary
condition is applied at
from the domain
=
to ensure the propagation of the scattered waves away
19
The parameter
analytic solution at
is the radiation coefficient which is modelled using the
=
vertical cylinder of radius
of the scattered velocity potential
of the fixed
as
where
and
In this expression,
and
are the Hankel function of the first kind and the
Bessel function of the first kind respectively,
= 0 and
, and
= 1 when
= 2 otherwise.
It is noteworthy that when deriving analytical solutions related to the OWC
boundary problem as in Evans & Porter (1997), it is often necessary to separate the
potentials from the scattering problem, related to the effect of the incident wave
potential, and the radiation problem, related to the effect of the oscillating pressure
inside the OWC chamber. The different waves possess distinctive physical and
mathematical properties as shown by Mei (1983) and each potential has to be
expanded into all its natural modes. Such expansion and separation is not required
when using numerical models based on FEM. In the FEM, the potential are
approximated within small elements using the shape functions and the solution can
include both radiated and scattered potentials. In the present study, the model was
directly applied to the overall boundary value problem.
20
The elements used in the present model are tetrahedrons with a node at each
vertex. The shape and test functions are polynomials of order one. The element sizes
were taken as invariable throughout the volume of the domain and were adjusted
depending on the frequency of the incident wave so as to have at least ten elements
per wavelength. The meshing was performed using the mesh generator included in
the ANSYS® software package. The mesh was then exported as an ASCII file into
MATLAB to apply the FEM discussed. Special attention was given to obtain a
sparse banded square stiffness matrix and the system of equations (2.22) was solved
using MATLAB banded linear solver.
The number of nodes and the number of elements were similar for each array
configuration, but they varied significantly depending on the wavelength of the
wave, from around 80000 nodes and 100000 elements in low
and 1000000 elements for high
to 170000 nodes
.
The model was run on the University of Wollongong (UOW) High
Performance Cluster (HPC) computing facility. The model itself did not require a
large RAM or data space and was usually run on one processor. The main advantage
of the HPC was the possibility to have up to fifty model runs simultaneously. The
CPU time for each frequency first run ranged from approximately 50 minutes for low
frequencies to around 130 minutes for high frequencies. Through the use of saved
pre-defined matrices from the first run, the CPU time for any additional runs at a
given frequency was substantially decreased. The CPU time for each frequency
additional run, usually related to the radiation problems, went from around 15
minutes for low frequencies to around 40 minutes for high frequencies.
2.4
Results and Discussion
2.4.1
Single Isolated OWC Device
A single isolated cylindrical OWC device was considered initially.
Dimensionless parameters defining the system were selected as follows: inner radius
= 0.2, outer radius
= 0.206 and the wetted depth of the cylinder
The computational domain had a maximum radius
of 98748 nodes and 567180 elements.
= 0.2.
= 25 with a mesh consisting
21
a
a)
)
b)
b
)
Figure 2.2: a) Dimensionless pneumatic damping coefficient
versus
(lower
axis) and corresponding
, (upper axis); b) Comparison of the efficiency
obtained using equation (2.16) and the efficiency obtained using the Evans & Porter
(1997) method.
A very thin OWC wall (
= 1.03) was chosen so that the FEM model could
be validated against the results of the method developed by Evans & Porter (1997).
The optimum damping coefficient, , was then computed for each frequency giving
so as to obtain the maximum output. The non-dimensional pneumatic
damping
is defined as:
22
where
is the density of the air. At 20°C at sea level this is approximately equal
to 1.2kg.m-3.
The frequency, at which maximum efficiency is attained, and the energy
bandwidth are strongly related to the physical dimensions of the problem. A practical
OWC system has to be properly designed according to the incident wave field and
topography of the selected wave energy device location. In the present case, the
maximum efficiency was found to be
Figure 2.b) at a wavelength
~ 0.75 corresponding to
~ 3.4 (see
~ 9.2.
In the method described by Evans & Porter (1997) the wall around the OWC
was considered to be infinitesimally thin. The efficiencies resulting from the present
model and from the method proposed by Evans & Porter (1997) agreed to within 1%
for
≤ 2 and
≥ 3.4. The effect of the finite wall thickness on the efficiency is
however clearly visible in Figure 2.2.b as a slight narrowing of the resonant peak as
compared with the efficiency calculated using the Evans & Porter method.
It is noteworthy that
is at a minimum at a frequency corresponding to the
maximum power output which in the present case is
~28 at
~3.4 (see Figure
2.a).
2.4.2
Arrays of OWC Devices
The model described in the previous section was then applied to the case of
multiple OWC devices. Two different array configurations were examined; i.e. a
four- and a nine-OWC array with the devices evenly spaced on a rectangular grid as
shown in Figure 2.3. Each device had the same dimension as the single isolated
OWC device (SIOD) as described in the previous section (
and
= 0.2,
= 0.206
= 0.2). The distance between two cylinders in the same row or the same
column was .
23
Figure 2.3: Schematic diagram of a) an array of 4 OWC devices, and b) an array of 9
OWC devices.
The OWC devices were all enclosed in the inner domain where the kinematic
boundary condition (10) was applied to their submerged surfaces. An oscillating
pressure was introduced inside the chamber of each OWC to represent the coupling
between the chamber turbine system that is used to generate electricity and the
motion of the water column inside the chamber. The pressures followed the relation
(14) relative to their own chamber volume flux and the boundary condition (15) was
applied on the surface of the water column inside of each chamber. The radiation
boundary condition (23) was also applied on the circular boundary
radiation coefficient
the value of
=
with the
being calculated from the expression (26) by replacing
in (26) with the radius
of a fictitious circle of the same perimeter as
that of the smallest square which enclosed the devices. It can be easily shown that the
radius
of such a circle is equal to
for an array of 4 OWCs
shown in Figure 2.3.a and
for the nine-OWC array shown in
Figure 2.3.b. Of course, the choice of the radius
was arbitrary. However it was
found that the variation between a radius from that of the inscribed circle and that of
the circumscribed circle of the square enclosing the array of OWCs was less than 1%
in the power output results from the model (see Table 2.1). The effectiveness of the
radiation boundary condition in this case was closely related to the distance of the
outer radius of the numerical domain
convergent results were found when
value of
.
from the array of OWCs (
>>
) and
was approximately five times greater than the
24
2
0.3344
0.3351
0.3345
3
0.4468
0.4501
0.4459
4
0.6619
0.6579
0.6653
5
0.1683
0.1695
0.1668
Table 2.1: Influence of the radiation boundary parameter
on the efficiency
obtained for the OWC device number 1 of the four device array configuration. The
angle of the incident wave is = π/4 and the array spacing is
= 10.
A regular monochromatic wave was applied to the model with an incident
angle
between the direction of wave propagation and the -axes. The model was
then executed for various spacing,
range of 1 ≤
and incident angles
over a wave frequency
≤ 6. The pneumatic damping coefficient for all OWCs was
considered equal to the optimum coefficient
for the SIOD as calculated using
the Evans & Porter (1997) method.
Two different efficiencies are defined for arrays of OWC devices. The mean
efficiency
where
is defined by the expression
is the efficiency as calculated in (2.17) and
is the mean power output
corresponding to the device number .
The capture width efficiency
is defined by the relationship
25
is the total array cross section width in the direction of the incident wave
front as presented in Figure 2.3 for the four-OWC array arrangement, and
is the
power capture width of the entire array system.
The efficiencies
and
reason as that for the efficiency
efficiency
can both potentially exceed unity for the same
.
and
are also of equal magnitude to the
for the case of the SIOD. The mean efficiency
gives a measure of
the total mean power output of an array of OWCs as compared to the mean power
output of a SIOD. On the other hand, the efficiency
gives a measure of the
absorption of the energy passing through the entire array system and can be directly
related to the power capture width of the whole arrangement related to the total
dimension of the array. The capture width efficiency is of particular interest when the
array is physically constructed as a single floating structure as is the case for a multichamber OWC device.
2.4.2.1 Mean Efficiency
This efficiency was computed for both arrangements with different values of
spacing
direction
= 1, 2, 5 and 10, with different angles of incident wave propagation
= 0, π/8 and π/4, through a frequency range of 1 ≤
≤ 6. The results of
the finite element modelling are presented in Figures 2.5, 2.6 and 2.7 .
It can be seen that for long wavelengths the power capture efficiency tends to
the corresponding SIOD efficiency. This is expected as the scattering interactions
between OWCs decreases as the wavelength, , becomes very large compared to the
dimensions
and . Each OWC therefore tends to behave as an isolated device. For
the same array spacing,
, both the four and nine device arrangements display a
similar performance although more complex behaviour with more defined peaks are
visible in the nine OWC array configuration mean efficiency plots. This is especially
apparent for the array spacings of
= 5 and 10.
An interesting and noteworthy outcome observed is that the mean power
capture efficiency of an array may be higher than the efficiency of the associated
SIOD. This result indicates that strategic placement of devices in arrays can
effectively increase the total power output.
26
The overall maximum mean efficiency for both the four and nine device arrays
is reached near the resonant frequency when the wave incidence angle is
when the spacing
= 0 and
= 5. This peak efficiency is up to 30% higher than the peak
efficiency attained with the SIOD. The spacing
= 5 effectively yields maximum
mean power capture efficiency when the wave incidence angle
= 0, but has the
worst performance of all array spacing values tested for frequencies
> 4. All other
cases trialled at this wave incidence angle generally produce lower maximum
efficiencies than the maximum efficiency of the SIOD. One notable exception is the
significant second peak with a maximum of
array with
For
~ 0.9 at
~ 3.7 for the nine OWC
= 10.
= π/8, the overall maximum efficiency is lower than for
= 0 but still
higher than the maximum SIOD efficiency by up to 21%. The most efficient
arrangement is when
= 10, whereas the other cases at this wave incidence angle
typically stay under the maximum efficiency of the SIOD. For the wave incidence
angle of
= π/4, the overall maximum is always lower than the maximum SIOD
efficiency. The incidence angle of
= π/4 can be considered to give the equivalent
of a diamond rather than a rectangular array grid. The results from the present study
indicate that a diamond array deployment configuration is clearly sub-optimal. The
arrangement with an incidence angle of
= 0, or deployment of a rectangular grid
array relative to the dominant wave direction for a particular site, seems to yield
significantly better power output.
27
a)
b)
c)
d)
Figure 2.4: Total wave amplitude around and in the four OWC device array for a
spacing of a)
= 1, b)
= 2, c)
= 5 and d)
= 10. The angle of the
incident wave is = 0 and the frequency is
= 3.4. The contour lines are plotted at
intervals such that
=0.1.
Arrays of OWC devices can be divided into three brand categories based on the
OWC spacing relative to the wavelength,
first category is where
, of the incident wave at resonance. The
. Within this region the arrays effectively behave as a
single, larger device with relatively low wave scattering present in the area between
the devices. Array arrangements with spacings of
= 1 and 2 can be categorised
in this way and the total wave field amplitudes are plotted around these arrangements
in Figure 2.4.a and Figure 2.4.b. The mean efficiency in such arrangements is found
to behave quite similarly and is generally inferior to the efficiency of the SIOD.
28
a)
b)
Figure 2.5: Comparison of the total mean efficiency
for array spacing of
= 1,
2, 5, 10 with the efficiency of the Single Isolated OWC Device (SIOD) versus
and corresponding
for = 0. a) corresponds to the four devices arrangements
and b) corresponds to the results obtained for the nine devices arrangements.
29
a)
b)
Figure 2.6: Comparison of the total mean efficiency
for array spacing of
= 1,
2, 5, 10 with the efficiency of the Single Isolated OWC Device (SIOD) versus
and corresponding
for = π/8. a) corresponds to the four devices arrangements
and b) corresponds to the results obtained for the nine devices arrangements.
30
a)
b)
Figure 2.7: Comparison of the total mean efficiency
for array spacing of
= 1,
2, 5, 10 with the efficiency of the Single Isolated OWC Device (SIOD) versus
and corresponding
for = π/4. a) corresponds to the four devices arrangements
and b) corresponds to the results obtained for the nine devices arrangements.
31
a)
b)
Figure 2.8: Comparison of the capture width efficiency
for array spacing of
=
1, 2, 5, 10 with the efficiency of the Single Isolated OWC Device (SIOD) versus
and corresponding
for = 0. a) corresponds to the four devices arrangements
and b) corresponds to the results obtained for the nine devices arrangements.
32
a)
b)
Figure 2.9: Comparison of the capture width efficiency
for array spacing of
=
1, 2, 5, 10 with the efficiency of the Single Isolated OWC Device (SIOD) versus
and corresponding
for = π/8. a) corresponds to the four devices arrangements
and b) corresponds to the results obtained for the nine devices arrangements.
33
a)
b)
Figure 2.10: Comparison of the capture width efficiency
for array spacing of
= 1, 2, 5, 10 with the efficiency of the Single Isolated OWC Device (SIOD) versus
and corresponding
for
= π/4. a) corresponds to the four devices
arrangements and b) corresponds to the results obtained for the nine devices
arrangements.
34
The second category of arrays is characterised by strong wave scattering
effects, and therefore strong interactions between devices. Such effects appear when
is of the order of and slightly larger than
as is the case for
= 5 and 10
(see Figure 2.4.c and Figure 2.4.d). The efficiency obtained in those cases is
therefore very dependent on the spacing
and multiple peaks appear throughout the
frequency range. A strategic selection of the array spacing in this category can
significantly enhance the total power output relative to a SIOD.
The third and final category of arrays is when the array spacing
large compared to
is very
. In this case the interaction between devices decreases
significantly and each device tends to behave as a SIOD. This category has not been
covered in the present study due to the very large mesh sizes required to model large
domains.
2.4.2.2 Capture Width Efficiency
The capture width efficiency was determined for both the four and nine device
array arrangements with spacings of
directions of
=1, 2, 5 and 10 and incident wave
= 0, π/8 and π/4 across the frequency range 1 ≤
≤ 6. The results
are presented in Figure 2.8, 2.9 and 2.10 .
It can be seen from these plots that the capture width efficiencies associated
with the smaller array spacing
decreases as
direction of
= 1 are the largest. For most cases the efficiency
increases. One minor exception may be seen for the incident wave
= 0, where the spacing
= 5 is slightly more efficient than
=1
around the resonant frequency. The nine-device array configuration generally
performs better than the four device arrangement. The different angles
seem to
have little effect on the capture width efficiency. The nine device arrangement at an
incidence angle of
= 0 clearly provides the best overall performance, achieving a
capture width efficiency up to 68% higher than for the SIOD. The maximum
efficiency of 1.15 occurs at
~ 3.3 for the case of
= 1. When considering the
capture width efficiency, the results of the present study appear to suggest that a
large number of chambers with a small spacing between them provide the preferred
array configuration. The aforementioned example of a single structure with multiple
35
OWC chambers has significant positive practical implications. The cost of such a
structure increases generally with size, both due to material and construction costs
and in dealing with the additional wave and current loading that the larger structure
experiences. Maximizing the power output capture width efficiency and reducing the
initial expenditure on a structure with a particular number of OWC chambers are
therefore complementary goals, both requiring minimal structure size.
2.5
Performance Optimisation
It can be seen that variations in the amplitude of the free surface within each of
the four OWC chambers of the array shown in Figure 2.4 are relatively small.
However, it can also be seen in Figure 2.4 that there are very significant differences
in mean displacement between the OWC chambers in the first and second rows of
the array and these differences are also found to be strongly dependent on the array
spacing. Similar differences are found in the air volume flux since it is directly
related to the integration of the free surface in each chamber (see equation (2.14)).
Hence there is a possibility that the optimal pneumatic damping coefficient for each
OWC chamber in an array may differ from the optimal pneumatic damping
coefficient for a SIOD.
To investigate this potential change in optimal damping and the effect of
position in the array, the case of the four device array configuration was selected
with a wave incidence angle of
= 0 and array spacing of
employing pneumatic damping coefficients of
on the efficiency,
2.5 ≤
= 5. The effect of
= 1, 0.5, 0.7, 0.9, 1.1, 1.3 and 1.5
, of each cylinder was examined over the frequency range of
≤ 4. Due to the symmetry of the arrangement at
= 0, only the results for
the OWC device number 1 and 2 are presented in Figure 2.11.a and 2.11.b,
respectively. In most cases
is observed to be very close to the optimal pneumatic
damping coefficient and
= 0.5 and 1.5 clearly appear sub-optimal. In the
frequency band near the maximum efficiency (i.e. 3.1 <
< 3.7) for the first row of
OWC devices, an improvement in the efficiency is possible by moving away from
the optimal pneumatic damping predicted for the SIOD. An increase in efficiency of
up to 5% is achievable with pneumatic damping less than
=0.5 and
between the values of
=0.9. Present results suggest that for practical purposes, optimal
36
damping values in the second row are unaffected by the presence of the other devices
within the array and that the optimal pneumatic damping coefficient in an array can
be different to the optimal pneumatic damping coefficient of a single isolated device.
Hence, this value is dependent on the position of the device within the array.
2.6
Conclusion
The aim of the present study is to contribute to the understanding of the
complex wave-structure interactions in arrays of fixed OWC devices. The FEM
model described in this chapter has shown that such interactions may be roughly
classified into three categories. These categories depend on the relative magnitude of
the wavelengths to the structure non-dimensionalised array spacing. When the
wavelength is large relative to the array spacing, then the array behaviour most
closely resembles that of a single large device. At the other extreme, when the
wavelength is small relative to the array spacing, the behaviour of the devices in the
array tends toward that predicted for a SIOD. In between these extremes (e.g.
=
5 and 10), the interaction effects are strong and the performance of the array and its
constituent devices are highly dependent on the spacing and location within the
configuration.
The results indicate that placement of the OWCs in an array can improve the
mean power capture efficiency. Improvements of up to 30% have been demonstrated
at an array spacing of
= 5. The trials conducted at various wave incidence angles
in the present study suggest that the array should be placed in alignment with the
dominant wave direction for maximum array efficiency. The poorest performance
was realised at a wave incidence angle of
= π/4. The capture width efficiency was
found to be generally higher for small array spacing configurations. This has
important practical consequences in the design of multiple-chamber devices.
37
a)
b)
Figure 2.11: Comparison of the efficiency
obtained for the OWC device a)
number 1 and b) number 2 of the four device array with pneumatic damping
coefficients of
= 1, 0.5, 0.7, 0.9, 1.1, 1.3 and 1.5 versus
and corresponding
. The angle of the incident wave is = 0 and the array spacing is
= 5.
38
The optimal pneumatic damping for individual OWC chambers in an array has
been shown to deviate from that predicted for a SIOD and can also vary for devices
within a given array. At an angle of wave incidence
= 0 for example, the front row
of OWC devices displays a lower optimal damping value in the frequency band near
maximum efficiency. The second row of devices showed no significant deviation
from the SIOD optimal damping. This again has practical implications for OWC
array design optimization.
The results presented in this chapter have demonstrated the complex nature of
the interactions within an array of OWC devices and their effects on the power
capture efficiency. In particular, the study has highlighted the need for thorough
array modelling prior to deployment of devices especially when large array spacing
is considered.
39
3 HYDRODYNAMIC AND ENERGETIC PROPERTIES OF A SINGLE
FIXED OSCILLATING WATER COLUMN WAVE ENERGY
CONVERTER
3.1
Introduction
The previous chapter presented the first application of the newly developed
FEM model to a single and a finite array of OWC devices. The study was, however,
limited in the description of the hydrodynamic properties of the system. Moreover,
following Evans & Porter (1997), the pneumatic damping coefficient was previously
considered positive and real. Such assumption overlooks the effect of air
compressibility inside the OWC chamber. As shown by Sarmento & Falcão (1985)
and, subsequently, by Martins-Rivas & Mei (2009a), (2009b), air compressibility can
have a non-negligible effect on the power extraction of an OWC device.
In this chapter, we start by considering a single fixed OWC device. We account
for air compressibility inside the chamber and a more in depth analysis of the
hydrodynamic properties is presented. It shows that only few specific hydrodynamic
coefficients are needed in order to study the dynamic and energetic behaviour of an
OWC device in waves. Using this method, it is also possible to directly derive the
closed form of the optimum damping coefficient so as to obtain maximum
hydrodynamic power extraction from the system. This method also provides the
foundation for the analysis of more complex systems as in the following chapters.
Finally, the method is applied to a cylindrical fixed OWC device with finite
wall thickness using a more efficient FEM model. Special attention is given to the
effect of air-compressibility and physical properties (draft, radius and wall thickness)
on the maximum hydrodynamic power available to the system.
3.2
3.2.1
Formulation
General Boundary Conditions
In this chapter, we consider a single fixed OWC wave energy converter
consisting of a truncated cylinder with a finite wall thickness. The cylinder is surface
piercing and operates in constant water depth
radius . The draft of the cylinder is
. The inner radius is
, the outer
. A Cartesian coordinate system
with
its corresponding cylindrical coordinates
are situated with the origin
40
coincident with the centre of the cylinder at the mean sea water level, the -direction
pointing vertically upwards as illustrated in Figure 3.1. Linear water-wave theory
with irrotational and inviscid flow is assumed. A monochromatic plane wave of
amplitude
and frequency
propagates from
. The computational
domain is separated into two regions with constant water depth ; an outer region
between a radius of
and
with a complex velocity potential
region, containing the OWC device, within the radius
potential
, and an inner
with a complex velocity
. Both potentials satisfy the Laplace equation
and
Figure 3.1: Schematic diagram of a single isolated OWC device
The velocity potential
velocity potential
and the velocity potential
wave by the device as
where
can be decomposed into the sum of the incident wave
induced by the scattering of the
41
In this expression,
equal to
is the gravitational constant and
, the parameter
is the wave number,
being the wavelength. The wave number
satisfies
the dispersion relation
A dynamic pressure
effect of the turbine, is assumed to oscillate inside the
OWC chamber at the same frequency as the incident wave and its complex value
can be introduced
The general boundary conditions for this problem are:

In the outer domain
:
On the sea floor at
On the surface, at
And the Sommerfeld radiation condition on
when
tends to infinity
42

Between the two regions at

In the inner region
:
:
At any point on the device walls
where
is the derivative in the direction of the unit vector
normal to the
surface of the wall and pointing outward of the fluid.
On the surface outside the OWC chamber, at
On the surface
where
3.2.2
created by the water at rest inside the chamber,
is the density of the water.
Turbine Characteristics and Dynamic Pressure
In contrast to the previous chapter, in the present analysis, air compressibility
inside the chamber is taken into account. Following Sarmento & Falcão (1985), the
relationship between the mass flow rate and the drop of pressure through a Wells
type turbine can be expressed as
43
where
is the rotational speed of turbine blades,
turbine rotor,
is the air density and
is the outer diameter of the
is the air volume inside the chamber.
is
an empirical positive coefficient which depends on the design, the number and set-up
of turbines. Under linear water wave theory with the assumption of small
perturbations, the mass flow rate can be approximated as
where
is the air density at atmospheric pressure and
the chamber at rest.
the air volume inside
is the total volume flux going through the turbine.
If we assume isentropic expansion or contraction then
being the velocity of sound in the air, and introduce the complex value
of the
total volume flux,
then the relation between the dynamic pressure and the volume flux, for a Wells type
turbine, can be expressed as
where
and
.
The coefficient of proportionality is, here, clearly a complex number. The real
part of the coefficient is related to the pressure drop through the turbine; whereas the
imaginary part of the coefficient represents the effect of the air compressibility inside
the chamber of the OWC device. It is physically understandable that air
compressibility will create a time-lag between the variation of the volume flux and
44
the variation of the pressure inside the chamber. This expression also suggests the
possibility of controlling, in real-time, the value of the pneumatic damping
coefficient,
by supervising the rotational speed of the turbine and, if possible, the chamber
volume of air.
The expression of the mean hydrodynamic power extracted by the system over
a wave period can be expressed as
where
represents the complex conjugate of
.
In practice, to obtain the effective mean power generated by the turbine, the
turbine performance curve needs to be taken into account. Moreover, losses due to
viscosity, generator and turbine speed maintenance also need to be subtracted from
the total power absorbed. However, there are many turbines with specific
characteristics as described in Curran & Folley (2008). The present chapter describes
a theoretical hydrodynamic analysis of OWCs. One of the key features of this
chapter and the following chapters is a detailed consideration of the issue of the
optimal parameters in order to achieve maximum hydrodynamic absorption. By
necessity significant portion of discussion and research has been devoted to quantify
these optimum parameters which is very important both theoretically and practically.
In fact turbines are usually designed depending on the hydrodynamic properties and
the coupling between the two comes as a second-stage in order to better tune the
system.
In this chapter, we are therefore mostly interested in the hydrodynamic part of the
system and the turbine performance curve and losses are not directly described.
45
For a fixed OWC device, the total volume flux going through the turbine is
equal to the volume flux induced by the oscillation of the free surface
inside the
chamber
By use of the expressions (3.18) and (3.21), the boundary condition (3.13)
becomes
For a given geometry, incident wave frequency
and turbine parameter
3.3
, chamber volume of air
, the system can now theoretically be solved.
Analysis
Under linear water wave theory, it is possible to separate the velocity potentials
into the diffracted wave potential
and the radiated wave potential
,
is related to the diffraction problem which describes the effects of the
interaction between the incident wave and the OWC device, in the case where no
pressure is present inside the device chamber. It solves the equations and boundary
conditions presented in Section 3.2.1 by simply considering the pressure
to be
-2
0N.m .
is related to the radiation problem which is induced by the forcing on the
water of the oscillating pressure
inside the chamber. It solves the same equations
presented in Section 3.2.1 by simply considering that
The volume flux
can then be separated into
0m2.s-1.
46
As a response of the incident wave amplitude,
can be considered
proportional to the incident wave amplitude
where
is the complex excitation coefficient.
Moreover, Falcão & Sarmento (1980) and Evans (1982) showed that the effect
on the water of the chamber pressure was analogous to replacing the free surface of
the chamber by an infinitesimally thin plate set oscillating.
considered proportional to the pressure
could therefore be
inside the chamber in the same way as
added mass and radiation damping are considered for a rigid-body system.
can be
expressed as
where
and
are real coefficients. Falnes & McIver (1985) named the coefficient
the radiation admittance with
the radiation conductance and
the
radiation susceptance, for the analogy with an electrical system. For a given
geometry, it was shown that the coefficients
frequency
.
,
and
are only dependent on the
would also be dependent on the direction of the incident wave
propagation if a non-axisymmetric device is considered.
It is important to remark, that such analysis is not restricted to a truncated
cylinder device in an open water environment, but can be applied, under linear water
wave theory, to any OWC geometry (e.g. Evans et al. (1995)) in any environment
(e.g. Martins-Rivas & Mei (2009a), (2009b)).
By introducing the expressions (3.25) and (3.26) into the relationship (3.18),
the dynamic pressure and the total volume flux inside the chamber can be expressed
as
47
and
The mean hydrodynamic power extracted becomes
and the mean power capture width can be defined as
This method is elegant in that only the coefficients
,
computed, for a given geometry and incident wave frequency
and
need to be
. Pressure, volume
flux, hydrodynamic power extraction and power capture width can then be directly
obtained from equations (3.27)-(3.30) for any desired parameters of the turbine
and chamber volume of air
.
In practice, for a typical Wells turbine, the volume of air
chamber cannot be easily changed and
inside the
is given by
Optimum hydrodynamic power extraction and optimum capture width can then
be obtained using the turbine parameters
48
leading to
and
In the situation where the turbines properties is able to exactly match the
hydrodynamic properties,
then, the well-known overall maximum hydrodynamic power (cf. Evans (1982)), for
this frequency, is reached
The maximum capture width becomes
From (3.35), it is clear that, for a typical Wells turbine, the maximum
hydrodynamic power can only be reached when
is negative. Other studies on air
turbines, such as Gato & Falcão (1989), Sarmento et al. (1990) and Gato et al.
(1991), have shown that the phase between the volume flux and the pressure can be
controlled by varying the rotor blades pitch angle of a modified Wells turbine and the
49
relation (3.35) could potentially be reached even for positive
. However, in order
to obtain a constant non-zero phase, stages appear in the cycle where the turbine has
to play the role of a compressor. It follows that even if the hydrodynamic maximum
power can be reached, power losses from the compressing stages can be quite
significant, decreasing the overall power output of the system. Gato et al. (1991) also
demonstrated that these losses are strongly dependent on the magnitude of the phase
shift between the volume flux and the pressure, and that it is usually only for small
phase shifts that the use of a modified Wells turbine becomes interesting.
In the following study, we only consider the maximum hydrodynamic power
for negative
and the relations (3.32)-(3.34) are used as maximum hydrodynamic
power for positive
with
= 0m4.s.kg-1. The power resulting of this method
represents the maximum hydrodynamic power available when a typical Wells turbine
is used as power-take-off. It also represents, for negative
, the maximum
hydrodynamic power available when a modified Wells turbine is in used. A modified
Wells turbine could, potentially, extract more efficient hydrodynamic power for
positive
3.4
but it is overlooked in this study.
The Finite Element Model
Figure 3.2: Example of a mesh used around the OWC device
The FEM used in this chapter is in methodology similar to the previous
chapter. However several areas of improvement were conducted. The use of
tetrahedron elements with a node at each vertex and at the middle of each edge and
quadratic shape functions were implemented. A thinner discretisation was used on
50
the free surface for higher precision and on the walls of the OWC device in order to
better shape the contours of the device using a quadratic approximation of the
geometry. The symmetry of the problem was also exploited by discretising only half
of the domain in order to decrease the number of elements (cf. Figure 3.2).
Moreover, tests on different radiation boundary conditions were furthermore
performed. It was found that even if the radiation condition, used previously in
Chapter 2, still gives accurate results, a second-order cylindrical damper, as
prescribed by Zienkiewicz et al. (2005), allowed a substantial decrease in the domain
size necessary to solve the problem. Convergence was found when the limit of the
domain was placed depending on the wavelength so as to have,
against
for the previous model. In this chapter, the following radiation
condition was, therefore, applied at the numerical limit of the model, at
, for
both the diffraction and radiation problem
where
is the tangential coordinate, at
and is equal to
.
A comparison between the models results and the analytical solution of a full
cylindrical cylinder was performed. Convergence under the 1% Root Mean Square
Error (RMSE) of the free surface was found using at least 10 elements per wave
length when using the previous model whereas a similar RMSE was obtained with
only 5 elements per wave length when using the new model. A RMSE of 0.32% was
then achieved with 10 elements per wave length.
The decrease in the domain size by use of the symmetry of the problem and the
new radiation condition decreased the number of elements and nodes necessary to
solve the system. The number of nodes and elements of the mesh varied from around
50000 nodes and 30000 elements in low
to 120000 nodes and 80000 elements for
51
high
against the 80000-170000 nodes and 100000-800000 elements ranges from
the previous model. Such decrease substantially improved the amount of CPU time
needed to run the model. The CPU time varied from approximately 30 minutes for
low frequencies to around 90 minutes for high frequencies and additional runs went
from around 5 minutes for low frequencies to around 20 minutes for high frequencies
against the 50-130 minutes first-run range and 15-40 minutes additional-run range in
the previous model.
Finally, from Section 3.3, it became clear that to study the dynamic and
energetic behaviour of an OWC device in wave, only the parameters
,
and
need to be computed, for a given geometry and incident wave frequency
. It
follows that the model only needs to be runt twice for each frequency. Firstly, the
model was applied to the diffraction problem only by considering the pressure
be 0 and the complex excitation coefficient
to
can be derived by computing the
volume flux inside the chamber
being 0 in this problem.
The model was then applied to the radiation problem only by considering the
incident potential
to be 0 and implementing an arbitrary chamber pressure
,
oscillating with the same frequency as the incoming wave in the related diffraction
problem. The radiation conductance
and the radiation susceptance
be computed by, once again, deriving the volume flux inside the chamber,
being 0 in this problem.
could then
52
3.5
3.5.1
Results and Discussion
Dimensionless Parameters
Prior to presenting and discussing the different results obtained from the
model, we shall introduce the non-dimensional form of the properties of interest.
The non-dimensional free surfaces are defined as
where ,
and
represent the total free surface, the diffracted-wave free surface
and the radiated-wave free surface, respectively.
The dimensionless radiation conductance, radiation susceptance, turbine
parameter and air compressibility parameter are chosen as
The dimensionless complex excitation coefficient is characterised by
which also represents the dimensionless volume flux of the diffraction problem,
.
The non-dimensional volume flux is symbolised by
And finally, the dimensionless capture width is represented by
53
a)
b)
a
.
b
.
c)
c
.
Figure 3.3: Wave amplitudes around and in the OWC device from a) the diffraction
problem
, b) the radiation problem
and c) the overall problem
. The
frequency is
= 3, and the parameters of the turbine and air compressibility are
and
.
3.5.2
Properties and Air Compressibility
In the following section, the dimensionless parameters defining the system
were selected as follows: inner radius
draft of the cylinder
= 0.2, outer radius
= 0.2. The model was applied to the diffraction problem
and the radiation problem for a set of frequencies between
After computing the parameters
pressure
= 0.25 and the
,
and
and
.
as described in Section 3.4, the
was derived from Expression (3.27) for any desired turbine parameter
and air chamber volume of air
. The model was reapplied to the radiation
problem by applying the calculated pressure
on the surface
of the OWC device. The total velocity potential
, inside the chamber
, for the overall problem at hand,
54
was computed by adding the diffracted wave velocity potential
wave velocity potential
and the radiated
. As a result, hydrodynamic properties (free surface,
pressure, velocity, etc.), all over the fluid, could be derived.
Figure 3.3 presents the different dimensionless free surface amplitudes around
and in the OWC device: the diffracted-wave free surface
radiated-wave free surface
(Figure 3.3.a), the
(Figure 3.3.b) and the reconstituted total free surface
(Figure 3.3.c). The frequency of the wave is
, and the parameter of the
turbine and air compressibility are
and
lines are plotted at intervals such that
. The contour
0.1. The volume flux for the
diffraction problem, presented in Figure 3.5, shows that
is situated somewhat
below the resonance frequency of the device although it is within the resonance
frequency band-width where the surface amplitude, inside the device, increases
significantly as the response to the incident wave potential (
chamber). The computed pressure, by use of the parameters
induced significant oscillations inside the OWC chamber (
inside the
and
, also
) but the
amplitude of the free surface decreases rapidly away from the device. The radiated
wave, when added to the diffracted wave, still has a considerable impact on the wave
field all around the device, as seen in Figure 3.3.c. At this frequency, the amplitude
inside the chamber seems to be dampened by the radiated wave going from
to
. The free surface inside the chamber is noticeably uniform
but a small gradient can still be observed in the diffracted and total free surface.
Accurate modelling away from the device is not necessary when focusing
mostly on the dynamic and energetic response of the system. It is, however, of major
concern when, as in the previous and following chapter, multiple devices are placed
in arrays and the interactions between the devices need to be rigorously modelled.
Figure 3.4 presents the dimensionless radiation conductance
susceptance
frequency
and radiation
computed by the model and plotted against the non-dimensional
.
is typically positive throughout the frequencies and reaches a
maximum of ~ 39.2 around
~ 3.21. In contrast,
frequencies, reaches a maximum of value ~ 18.4 at
the same frequency of the
is positive in the lower
~ 3.05, changes sign around
maximum, and passes through a minimum of ~ -21 at
~ 3.4. Such behaviour is in agreement with the results presented by Evans &
55
Porter (1997) and Falnes & McIver (1985) for a fixed cylindrical OWC device with
infinitely thin wall. In Figure 3.5,
~ 3.21 also aligns with the maximum volume
flux amplitude for the diffraction problem. For this configuration,
~ 3.21
characterised the natural resonance frequency of the OWC device.
Figure 3.4: Dimensionless radiation conductance , radiation susceptance
compressibility parameter
for different chamber volume of air,
6.82 and 15
The -axis represents the dimensionless frequency .
and air
0, 1, 3,
In this part of the study, we are looking at the effect of air compressibility on
the hydrodynamic and energetic behaviour of the OWC device. Figure 3.4 also
presents the value of the negative air compressibility parameters
chamber volumes of air,
0, 1, 3, 6.82 and 15
, for different
As discussed in Section 3.3,
for a typical Wells turbine, overall maximum hydrodynamic power can be extracted
when
. This relation can only be satisfied for negative
, which, for this
problem, only happens for frequencies higher than the natural frequency of the
system. For this specific geometry, it was found that this equation has one solution at
the resonance frequency of the device, when
3.4
when
~
6.82
.
= 0m3 and at the frequency
has
two
solutions
when
, where the lowest frequency is situated between the
56
resonance frequency
3.21 and
3.4, and the second frequency can be
situated anywhere in the segment
depending on the value of
relation is, however, never satisfied when
> 6.82
.
Figure 3.5 shows the total dimensionless volume flux
using the optimum turbine parameter
0, 1, 3, 6.82 and 15
non-dimensional volume flux
volume flux
plotted against
, for the different chamber volumes of air,
These volume fluxes are also compared with the
related to the diffraction problem and the total
using the maximum turbine parameter
compressibility parameter
. The
and air
. The behaviour of the optimal volume fluxes can be
separated into two frequency band-zones, before and after the natural resonance
frequency of the OWC device or more accurately depending on the sign of
.
For low frequencies, the amplitudes of the different volume fluxes are typically
m3 and increase toward
higher than the amplitude of the volume flux for
as the volume of air increases. From Expression (3.32), as
towards infinity,
and
will increase towards infinity leading the pressure
, (from Equation (3.27)), to tend to 0N.m-2. As a result,
for large chamber volumes of air. The values of
between 0.71 for
increases
0m3 and 1 for
represents the limit of
, for low
. As
, are comprised
increases, the different
amplitudes rise toward a maximum at the natural resonance frequency of the OWC
0m3 and 3.2
device. These maxima are comprised between the values 2.83 for
for
. For frequencies below the resonance frequency,
volume flux associated with the parameters
volume flux associated with
and
is positive and the
is identical to the
0m3.
For frequencies higher than the natural frequency of the OWC device, a
completely different behaviour is noticeable.
and 15
, related to
1, 3, 6.82
, exhibits a second maximum, slightly lower than the first peak. The
position of these peaks is dependent on
; the peak is situated close to the natural
resonance frequency of the device for the largest volume of air tested,
and is shifted towards higher frequencies for lower
associated with
.
and
m3 do not exhibit a second peak. Conceptually, the two
peaks can be seen as merged together for
whereas for
associated with
57
0m3, the frequency of the second peak can be thought of as pushed to infinity.
But one of the most remarkable effects of the added phase between the volume flux
and the pressure, induced by the air compressibility, is the fact that
greater than
can become
. The pressure, instead of having a dampening effect, as for the
lower frequencies, can have an exciting effect on the amplitude of the volume flux.
associated with the parameters
and
does not exhibit a second
maximum but displays slowly decreasing values from its maximum.
becomes higher than
when the relation
, and is equal to the different
is verified.
particular position in the behaviour of the related
also
related to the fixed
does not seem to represent any
except that they are placed
between the two maxima.
Figure 3.5: Non-dimensional amplitudes, versus , of the diffracted wave volume
flux
and the total volume flux
using the turbine parameter
for
different chamber volumes of air,
0, 1, 3, 6.82, 15
and using the
parameters
and
.
58
Figure 3.6: Dimensionless optimum capture width
for different chamber
volumes of air
0, 1, 3, 6.82, 15
and dimensionless maximum capture
width
for different .
Finally, Figure 3.6 presents the results obtained in terms of dimensionless
optimum capture width
6.82, 15
for different chamber volumes of air
plotted against
maximum capture width
0, 1, 3,
and are compared with the dimensionless
, as described in Section 3.3. As expected,
always higher or equal to the different
. The behaviour of the
is
can also
be separated into two frequency zones, before and after the natural resonance
frequency of the OWC device.
In the low frequency area, all the dimensionless capture widths tend to zero for
small
and increase as the frequency increases. In this section,
therefore equals
of
associated to
is positive and
0m3. For a given
, The value
typically decreases as the volume of air inside the chamber increases. Then
the overall maximum capture width is obtained by
natural frequency of the OWC device,
area, the behaviours of the
, slightly before the
~ 3.2, of around 0.8. In the high frequency
become more complex. All the
maximum around the natural resonance frequency of the OWC device.
exhibit a
~ 0.8,
59
0.798, 0.792, 0.777, 0.67, at
15
~ 3.2, 3.22, 3.23, 3.28, 3.28 for
0, 1, 3, 6.82,
, respectively. The capture width corresponding to
15
stays
suboptimum at all the frequencies tested, displaying a small band-width and a single
peak. For
0, 1, 3, 6.82
, the maxima around the resonance frequency of
the device, appear closely before the respected relations
peak of the capture width associated with
6.82
are verified. The
, is quite large around the
maximum but
decreases quickly before and after the peak. The capture
widths related to
1 and 3
exhibit a second peak with maxima of
~
0.667, 0.488, appearing, once again, shortly before the second solution of the relation
is verified at
~ 3.9, 5.38. The apparition of these second peaks has also
for effect to increase significantly the power capture frequency band-width of the
system. Finally, for high frequencies each of the
tends to zero and seems to
tend to be in the same order as for low frequencies (highest values for small
smallest values for large
). In the high frequency zone,
significantly higher than the different
could, however, picture
is generally,
with slowly decreasing values. We
to coincide with the maxima of
volume of air changes from 0m3 to 6.82
and
as the
.
From Figure 3.5 and Figure 3.6, it is evident that high volume flux amplitudes
do not necessarily correspond to high capture widths. Moreover, for a given volume
flux amplitude, a range of capture widths can result. Overall maximum
hydrodynamic power extraction is found when the turbine parameter
compressibility parameter
and the air
, most closely match the hydrodynamic properties of the
OWC device.
From this study, we can clearly observe the non-negligible effects of the air
compressibility induced by the volume of air inside the chamber. A careless choice
of air volume could considerably diminish the overall hydrodynamic power
absorption whereas a carefully chosen volume of air could considerably increase the
overall capture band-width of the system. It follows that the chamber volume of air is
an important physical parameter in the design of OWC devices.
60
a)
b)
Figure 3.7: Dimensionless a) radiation conductances
versus
for different OWC device draft
0.2 and
0.25.
and b) radiation susceptances
0.1, 0.15, 0.2, 0.3 and 0.5.
61
a)
b)
Figure 3.8: Dimensionless a) amplitudes of the complex excitation coefficient
and b) maximum capture widths
versus
for different OWC device draft
0.1, 0.15, 0.2, 0.3 and 0.5.
0.2 and
0.25.
62
3.5.3
Physical Properties Effects
In this section, effects on the dynamic and energetic behaviour of the OWC
device, due to its geometrical properties, draft
, inner radius
and wall thickness
, are studied. In each of the figures, Figure 3.7, 3.8, 3.9, 3.10, 3.11 and 3.12
the radiation conductances
, the radiation susceptances
complex excitation coefficient
versus the dimensionless frequency
, the amplitudes of the
and the maximum capture widths
,
, are presented with one of the physical
parameters varying while the others are fixed.
represents the maximum
hydrodynamic power available to the system when a typical Wells turbine is used as
power take-off. In view of the results presented in the previous section (Section
3.5.2), and by use of the radiation susceptances
and maximum capture widths
, we could predict what the effects of air compressibility would be on the
extracted power of the system.
was, therefore, chosen as an indication of the
energetic behaviour of the system so as to liberate ourselves to the dependency on
when comparing the hydrodynamic power absorbed.
3.5.3.1 Draft
Figure 3.7 and 3.8 presents the results computed by varying the draft of the
cylinder
0.1, 0.15, 0.2, 0.3 and 0.5, through the dimensionless frequencies 0.1
6, while the inner and outer radii are kept fixed,
0.2 and
0.25.
The first obvious effect is an inverse relationship between the draft length and the
position of the natural resonance frequency of the device. The resonance frequency
of the device is found at
~ 4.87, 3.87, 3.21, 2.44 and 1.71 for
0.1, 0.15,
0.2, 0.3 and 0.5, respectively. As the draft of the OWC increases, each of the
parameters,
,
and
, shows an amplification in their maximum and minimum
values but with a clear narrowing in each of the peaks.
behaviour than the other parameters,
Following the same
associated with the longest draft tested,
0.5, displays an overall maximum, around its natural resonance frequency,
considerably higher than the other drafts tested,
maxima decrease as the draft shortens:
1.52. The capture widths
1.08, 0.8, 0.68 and 0.53 for
0.15, 0.2, 0.3 and 0.5, respectively. However, as
increases after the different
63
maxima, the longer the draft, the steeper the rate of decrease in
, the capture width associated with
compressibility, the relation
and for large
0.5 is the lowest. In terms of air
, related to
0.5, can be solved by a
broader region of values for the chamber volume of air, but most of the air volumes
will lead to two solutions very close to each other on the frequency axis and it can be
shown that the power capture band-width related to these solutions will be very
narrow. Only for small values of
will it be possible to increase the power band-
width. The other drafts, even if having lower capture width maxima, and allowing
slightly narrower region of values for
larger choice of
, will allow wider power band-widths for a
.
3.5.3.2 Inner radius
Figure 3.9 and 3.10 displays the dynamic and energetic behaviour of the OWC
device as in Figure 3.7 and 3.8. Variable inner radii,
0.1, 0.15, 0.2,
0.25 and 0.3, are studied through the dimensionless frequencies 0.1
6, while
the draft of the cylinder and the wall thickness are held constant,
0.2 and
0.05. Similarly to changing the draft, changing the radius of the
cylinder induces a shift in the natural resonance frequency of the device. The
resonance frequency of the device is found at
~ 3.84, 3.5, 3.21, 3 and 2.83 for
0.1, 0.15, 0.2, 0.25 and 0.3, respectively. An amplification in maxima and
minimums of the parameters,
,
and
, and a narrowing in the different peaks
are also noticeable for decreasing values of . Overall maximum capture width of
1.35 is found for
0.1. A similar analysis for the change in inner
radius than for the change of draft could be carried out, though the narrowing of the
peaks seems to be comparatively less pronounced than previously.
64
a)
b)
Figure 3.9: Dimensionless a) radiation conductances and b) radiation susceptances
versus
for different OWC device inner radius
0.1, 0.15, 0.2,
0.25 and 0.3.
0.2 and
0.05.
65
a)
b)
Figure 3.10: Dimensionless a) amplitudes of the complex excitation coefficient
and b) maximum capture widths
versus
for different OWC device inner
radius
0.1, 0.15, 0.2, 0.25 and 0.3.
0.2 and
0.05.
66
3.5.3.3 Wall Thickness
Earlier studies on OWC devices such as Evans & Porter (1997) or Falnes &
McIver (1985) often considered infinitesimally thin walls. However, when the device
is placed in an open-water environment, even if fixed through the installation of
moorings or mounted on pedestals, the addition of buoyancy to the structural system
would most likely be necessary. Finite-wall thickness, as in this study, can be used to
take account of this buoyancy requirement. Figure 3.11 and 3.12 displays the
dynamic and energetic behaviour of the device, for dimensionless frequencies 0
6, for different wall thicknesses. Wall thickness was determined by varying
the outer radius , such that
0.21, 0.25, 0.3 and 0.4, while the inner radius and
draft of the cylinder were kept unchanged,
0.2 and
A more subtle shift in the natural frequency,
observable as the outer radius
increases,
0.2.
3.34, 3.21, 3.14 and 3.04, is
0.21, 0.25, 0.3 and 0.4,
respectively. An amplification in maxima and minima of the parameters,
, and a narrowing in their peaks can also be observed as
effect on the capture width
,
and
increases but the
is more subtle. In the low frequency zone, higher
capture width is found for lower values of
but as
increases, the curves exhibit
steeper slopes before reaching their maxima. The differences in maximum value are
not as large as previously,
0.76, 0.8, 0.81 and 0.83 for
0.21, 0.25,
0.3 and 0.4, respectively. Then, each of the curves display a similar decrease of the
slope as
increases, however, highest
seems to be found for
0.25.
67
a)
b)
Figure 3.11: Dimensionless a) radiation conductances
and b) radiation
susceptances
versus
for different OWC device outer radius
0.21, 0.25,
0.3 and 0.4.
0.2 and
0.2.
68
a)
b)
Figure 3.12: Dimensionless a) amplitudes of the complex excitation coefficient
and b) maximum capture widths
versus
for different OWC device outer
radius
0.21, 0.25, 0.3 and 0.4.
0.2 and
0.2.
69
3.6
Conclusion
For the study the dynamic and energetic behaviour of a single fixed OWC
device, the method presented in this chapter is particularly efficient as only the
hydrodynamic parameters
,
and
and incident wave frequency
need to be computed, for a given geometry
. Pressures, volume fluxes, power extractions and
power capture widths can then be directly obtained from equations (3.27)-(3.30) for
any desired parameter of the turbine
turbine parameter
and chamber volume of air
. Optimum
, leading to maximum hydrodynamic power extraction
and maximum capture width
(3.32) to (3.34) for a given
, can also be directly computed from equations
.
Using this method, the FEM model was applied to the study of the dynamic
and energetic behaviour of a cylindrical OWC device with finite wall thickness. The
model was executed only twice for each frequency, once for the diffraction problem
and once for the radiation problem, as described in Section 3.4. The different
hydrodynamic coefficients could then be derived through the computation of the
different volume fluxes. The study especially focused on the consequences of
changing the device’s geometrical properties. It was observed that due to air
compressibility, the chamber volume of air
had significant effects on the
dynamic and energetic performance of the device. A careless choice of air volume
can considerably diminish the overall hydrodynamic power absorption whereas a
carefully chosen volume of air can considerably increase the overall capture bandwidth of the system. The choice of
system. Influences of the change in draft
is therefore an important parameter of the
, inner radius
and wall thickness (
)
are also studied. It is found that each of these parameters have a large effect on the
position of the natural frequency of the system, on the amount of hydrodynamic
power extraction and on the frequency band-width.
Design, prior to installation of an OWC device, is therefore decisive. Each of
the physical parameters studied can be specifically devised as to best fit the energetic
behaviour of the system in connection with the local wave climate of the
emplacement of interest.
70
4 HYDRODYNAMIC AND ENERGETIC PROPERTIES OF A FINITE
ARRAY OF FIXED OSCILLATING WATER COLUMN WAVE ENERGY
CONVERTERS
4.1
Introduction
The present chapter naturally follows the preceding two chapters. Chapter 2
presented one of the first studies using the newly developed FEM model. It
demonstrated the complexity of the interactions between OWC devices within finite
arrays of these systems. Investigation of the pneumatic damping coefficient showed
that optimum coefficients for each of the devices might differ as for an isolated
device.
Chapter 3 contained a more in-depth analysis of the hydrodynamic properties
of a single OWC device and took into account the effect of air compressibility. One
of the advantages of this approach was the possibility to derive the closed form of the
optimum turbine parameters for maximum hydrodynamic power extraction by the
device.
Falnes & McIver (1985) introduced the theory of the interaction between
oscillating systems. In the present chapter, we apply the theory to a finite array of
fixed OWC devices. In contrast to Falnes & McIver (1985), the coupling between the
pressures and the volume fluxes are expressed by considering that a Wells type
turbine is used as the power take-off system. The interactions between devices and
air compressibility are taken into account and a new method for turbine optimisation
is developed. The method is then applied to three different arrangements of OWC
devices using the 3D FEM model.
An infinite periodic array of OWC devices were studied by Falcão (2002), but,
to the present author’s knowledge, direct application of the hydrodynamic method
developed in this chapter to an explicit study of the dynamic and energetic behaviour
of a finite array of fixed OWC devices, has not yet been published.
4.2
4.2.1
Formulation
Boundary Conditions
We start by considering a system consisting of a number
of fixed OWC wave
energy converters of arbitrary shape. The OWC devices are randomly numbers from
71
1 to . Linear water-wave theory with irrotational and inviscid flow is assumed. A
monochromatic plane wave of amplitude
and frequency
propagates from
. The computational domain is separated into two regions with constant
water depth ; an outer region between a radius of
and
with a complex velocity
potential
, and an inner region within the radius
with a complex velocity
potential
. The inner region is considered to contain all the OWC devices. Both
potentials satisfy the Laplace equation
and
The velocity potential
velocity potential
can be decomposed into the sum of the incident wave
and the velocity potential
induced by the scattering of the
wave by the array,
where
In this expression,
is the gravitational constant and , the wave number, is
equal to 2π/ . The parameter
is the wavelength. The wave number
satisfies the
dispersion relation
In order to model the effect of the turbine, dynamic pressures, inside each of
the OWC chambers, are assumed to oscillate at the same frequency as the incident
72
wave. The complex values of these pressures can be introduced and the pressure
inside the OWC device number can be expressed as
The general boundary conditions for this problem are:

In the outer domain
:
On the sea floor at
On the surface, at
And the Sommerfeld radiation condition on

Between the two regions at

In the inner region
At any point on the devices walls
:
:
when
tends to infinity
73
where
is the derivative in the direction of the unit vector
normal to the
surface of the walls and pointing outward of the fluid.
On the surface outside the OWC chambers, at
And on the surface
, created by the water at rest inside the chamber of the OWC
device number ,
where
4.2.2
is the density of the water.
Turbine Characteristics and Dynamic Pressures
In the same way as in the previous chapter, the dynamic pressures inside each
of the OWC devices are considered to be proportional to their respective volume
fluxes. By considering air compressibility, as in Sarmento & Falcão (1985), the
relationship between the dynamic pressure
and volume flux
, inside the
chamber of the OWC device can be expressed as
where
and
is the air density at atmospheric pressure,
the air,
is the rotational speed of turbine blades,
turbine rotor and
.
is the velocity of sound in
is the outer diameter of
is the air volume inside the chamber at rest.
is an empirical
positive coefficient which depends on the design, the number and set-up of turbines.
The devices being fixed,
can be expressed as
74
The boundary condition (4.13) becomes
For a given geometry, incident wave frequency
turbine parameters
4.3
, volumes of air
and
the system can now theoretically be solved.
Analysis
In this problem and under linear water wave theory, it is possible to separate
the velocity potential into
is the diffracted wave potential and represents the effects of the scattering
induced by the incident wave potential with the array of OWC devices in the case
where no dynamic pressure is present inside any of the devices’ chambers.
are the radiated potentials related to the effects of the forcing on the water
by each of the oscillating pressures
separetly. The volume flux
, inside the
device number , can then be separated into
As in the previous chapter,
wave amplitude
can be considered proportional to the incident
75
where
is the excitation coefficient, inside the OWC chamber . Following the
theory developed by Falnes & McIver (1985), the volume fluxes
OWC device
inside the
can be considered to be proportional to the respective pressures
giving
where
,
and
are, respectively, the radiation admittances,
the radiation conductances and the radiation susceptances contributed by the
pressures
to the volume flux
The coefficients,
inside the chamber of the OWC device .
,
and
dependent on the incident wave frequency
are, for a given geometry, only
.
would also be dependent on the
direction of the incident wave propagation and the -axes, if considered.
Introducing the different expressions for the volume fluxes into the expression
(4.18), we can derive the relationship
where
being the Kronecker delta.
equals one if equals to
and zero otherwise.
By considering the expression (4.21) for each of the OWC devices, we obtain
the system of equations represented in its matrix form by
76
where
The coefficients
and
are strongly dependent on the geometry of the
system and on the shapes of the OWC devices. However, due to reciprocity
influences, it could be shown that
The matrix
is therefore symmetrical. The determinant of the matrix
dependent on the parameters coefficients
related to the different turbines
properties. In the following, we can simply consider
the determinant of the matrix
to be chosen in order that
is not zero. It follows that for the OWC device ,
can be expressed as the product of the incident wave amplitude
result of the system (4.23).
of the matrix
is
and a function
is a specific arrangement of the different coefficients
and the complex excitation coefficients
. As a consequence,
is only dependent, for a given geometry, on the incident wave frequency
, and each
of the turbine parameter coefficients
The mean hydrodynamic power
can then be expressed as
, over a wave period, and capture width
,
77
and
for the OWC device l.
We can see from this method that the complex excitation coefficients
radiation conductances
and the radiation susceptances
are the key
parameters of the system. For a given geometry and incident wave frequency
pressures
, volume fluxes
, power extractions
, the
, the
and power capture widths
can then be directly obtained from (2.18)-(2.27) for any desired turbines
parameters
and chamber volumes of air
.
In the following, three different cases of arrays of OWC devices will be
presented. One of the purposes of these studies is to obtain, for each frequency, the
optimum turbine parameters
in order to maximise the overall power extraction
of the system. In order to achieve this optimisation, we will be considering the total
mean capture width of the system
becoming
when optimised.
As in Chapter 3 we will also be considering the overall maximum mean
hydrodynamic power available to the system when typical Wells turbine are
considered for power take-off. Such optimisation can be performed on the mean
capture width of the system by considering the parameters
to be also variable.
78
will, however, be considered to be positive or zero, and the optimised results of
this method will be called
4.4
,
and
.
The Finite Element Model
b)
a)
c)
Figure 4.1: Examples of the meshes used around the OWC devices for the different
problems studied in Section 4.5. a) A column of two OWC devices, b) A row of two
OWC devices and c) Two rows and two columns of OWC devices. These meshes are
related to the non-dimensional frequency
.
In the following, the application of the method to three different arrangements
of arrays using the 3D FEM model is presented. The FEM model used was very
similar to that in the previous chapter, the main difference being the implementation
of multiple pressures inside the OWC chambers. The number of nodes, the number
of elements as well as the CPU time for each case were also very similar to the single
OWC problem from Chapter 3. Figure 4.1 presents examples of the meshes, for
3, used for the different problem studied in Section 4.5. After further testing
the second-order cylindrical damper was, once more, considered for this problem.
At
,
79
where
is the tangential coordinate, at
and is equal to
function of the wavelength, insuring that
.
was placed in
.
In order to derive the different complex excitation coefficients
radiation conductances
and the radiation susceptances
, the
necessary to study
the dynamics of the system, the model was applied to the diffraction problem and the
different radiation problems, separately. From the diffraction problem, the complex
excitation coefficients
could be derived by computing the volume flux inside the
chamber of each of the OWC devices
for the device number , all the
being equal to 0 in this problem.
The model was then applied to each of the radiation problems by
implementing an arbitrary chamber pressure
, only in the chamber of the device ,
oscillating with the same frequency as the incoming wave in the related diffraction
problem. The radiation conductances
and the radiation susceptances
could then be computed by deriving the volume flux inside each of the chambers of
the different devices in the array
the different
being equal to 0 in these problems.
80
Following the processing of the complex excitation coefficients and the
radiation admittances, each of the pressures
turbines parameters
can be derived for any desired
and air chambers volumes of air
. The model can then
be reapplied to the different radiation problems with the appropriate pressure. The
resulting velocity potentials can finally be summed with the diffracted wave potential
to obtain the final velocity potential of the general problem.
4.5
Application
4.5.1
Dimensionless Parameters
Prior to presenting and discussing the different problems to which the model
was applied, we introduce the non-dimensional form of the properties of interest.
The non-dimensional free surfaces are defined as
where
,
,
and
represent the total free surface, the free surface
corresponding to the diffraction problem, the free surfaces corresponding to the
radiation problem induced by the pressure
separately, and the free surface
corresponding to the sum of all the radiation problem,
, respectively.
The dimensionless radiation conductances, radiation susceptances, turbine
parameters and air compressibility parameters are chosen as
where
.
The complex excitation coefficients are non-dimensionalised as follow
81
which also represents the dimensionless volume fluxes of the diffraction problem,
.
The non-dimensional volume fluxes are symbolised by
And finally, the dimensionless capture width are characterised by
where
4.5.2
.
A Column of Two OWC Devices
4.5.2.1 Analysis
1
2
Figure 4.2: Schematic Diagram of the arrangement of a column of two identical,
fixed, cylindrical OWC devices.
In this section, we are considering an array of two identical, fixed, cylindrical
OWC devices with finite wall thickness. The devices are set in a column
arrangement and numbered as illustrated in Figure 4.2. The OWC inner radius is
82
= 0.2, the outer radius of the wall is
= 0.25 and the draught
spacing between the two OWC devices is chosen to be
= 0.2. The
= 5.
Due to symmetry of the diffraction and the radiation problems, and considering
the reciprocity relations (4.25), the following equalities can be deduced
If we consider that the devices possess the same chamber volume of air
,
we can presume that the optimum turbine parameters, and therefore the dynamic
pressures, the volume fluxes, and power extractions for the two OWC devices are the
same,
Solving equation (4.23) for this problem, and taking into accounts the
relations (4.39) and (4.40), it can be shown that the pressures for both the devices can
be expressed as follow
The capture widths for each of the device are also equal, in this problem, to
the total mean capture width,
83
These expressions are noticeably similar to the Expressions (3.28) and (3.29),
from Chapter 3, of a Single Isolated OWC Device (SIOD). The only difference is the
presence of the radiation conductance
to
and
and radiation susceptance
, added
. These new components reflect on the radiation influences
between the two devices.
The optimum total mean capture width can then be obtained by modifying
Expression (3.32) as follow
giving
Moreover, the maximum total mean hydrodynamic power available can be
directly obtained by using
when
or
is negative or zero, giving
84
when
is positive, giving
a)
b)
c)
d)
Figure 4.3: Dimensionless free surface amplitudes attached to a) the radiation
problem 1,
, b) the sum of the radiation problems,
, c) the diffraction
problem
, and d) the overall problem,
. The frequency is
= 3, and the
parameters of the turbine and air compressibility are
and
.
85
4.5.2.2 Results and Discussions
Following the method described in Section 4.5, the values of the radiation
conductances
and
, the radiation susceptances
complex excitation coefficient
and
, and the
were computed and are presented, in their
dimensionless form, in Figure 4.4 and 4.5. Moreover, after deriving the pressure
from expression (4.41), using the parameters of the turbine and air compressibility
and
, the model can be reapplied to the radiation problem with the
appropriate pressure.
The different free surfaces can also be computed and, as an example, Figure
4.3 presents the resulting dimensionless free surface amplitudes around and in the
OWC device for an incident wave frequency of
surface amplitude
is presented in Figure 4.3.a, the sum of the radiated wave
free surface amplitudes
surface
. The radiated wave free
is presented in Figure 4.3.b, the diffracted wave free
is presented in Figure 4.3.c and the reconstituted total free surface
is presented in Figure 4.3.d. The contour lines are plotted at intervals such that
0.1. From Figure 4.3.a, we can observe that the radiation problem is not
axisymmetric as it was for the SIOD. We can also clearly see the response amplitude
in device 2 which is induced by the pressure inside the device 1. This amplitude is
higher than the amplitude directly outside of the device. That can be explained by the
fact that
= 3 is close to the resonance frequency of the system where a small
stimulus can lead to a large response in amplitude. The sum of the radiated wave free
surfaces demonstrates the interactions between the different radiated waves and,
when added to the diffracted wave, it has a substantial impact on the total wave field
environment all around the devices. This result is particularly evident when looking
at the differences between Figure 4.3.c and 4.3.d.
Visible in Figure 4.4, the radiation conductance
susceptance
and the radiation
are fairly similar compared to the radiation conductance
radiation susceptance
and
of the SIOD. Small differences are nonetheless visible
around the natural resonance frequency of the device. These discrepancies are of
great importance as they demonstrate that the inner properties of a device can change
depending on whether it is isolated or placed in an array. Moreover, the radiation
86
conductance
and radiation susceptance
exhibit non-negligible values
around the resonance frequency of the device. Such results demonstrate the
significant influence that the radiated wave, induced by the pressure of other devices,
can have on the hydrodynamic properties of the system. From the behaviour of
and
SIOD,
, we can observe that the resonance frequency is still the same as for the
~ 3.21.
and
display comparable behaviour to
but with smaller amplitudes. It is noteworthy that
negative values. In the same way as
changing of sign of
and
, but not
, the maximum of
characterise the resonance frequency
and
, displays
and the
~ 3.21. This study
is a particular case where the two devices are identical. It could be shown that if the
two devices were different,
and
would characterise the resonance
frequencies of both the devices.
Figure 4.4: Radiation conductances
and
and radiation susceptances
and
compared with the radiation conductance
and radiation susceptance
of the SIOD, versus .
87
The amplitude of the complex excitation coefficient
shown in Figure 4.5
is significantly higher than for the SIOD. From Figures 4.5 and 4.6, we can observe
that air compressibility influences both the amplitudes of the volume flux
the optimum capture widths
and
in the same way as discussed in the previous
chapter for the SIOD.
Figure 4.5: Amplitude of the dimensionless complex excitation coefficients
compared with the one from the SIOD, and amplitude of dimensionless volume flux
for different turbine and compressibility parameters, and versus .
As seen in Figure 4.6, and as previously discussed in Chapter 2, devices in
arrays can extract more hydrodynamic power than the same number of SIODs. For
low frequencies,
related to the array is slightly higher than
related to
the SIOD. It, however, displays a significant increase in maximum capture width
around the natural resonance frequency. For the array,
0.8 for the SIOD. In the high frequency domain,
0.96 compared to
related to the
array behaves quite differently than for the SIOD. Instead of exhibiting slowing
decreasing values, it passes through a minimum around
4.7, with values lower
88
than the SIOD, before increasing as
increases with values becoming, once more,
greater than the SIOD.
Figure 4.6: Dimensionless optimum capture width
for different air chamber
volume of air and maximum dimensionless capture
compared with the
of the SIOD and the capture width
obtained with the parameters
and
versus .
In Figure 4.6, we also compare the maximum capture
width.
.
with the capture
represents the power extraction of the device using the method
applied previously to the maximum power extraction of the SIOD problem and it
represents the case where the interaction between the devices are disregarded.
Depending on the sign of
of air compressibility
If
otherwise
, the parameter of the turbines
related to
are chosen as follow:
and the parameter
89
The capture width
, displays similar values to
for the lower
frequencies. However, from the natural resonance frequency onward, it exhibits an
almost constant 0.1 lower value than
. This difference might not seem of
great importance, but multiplied by the power of the incident wave, the number of
devices and time, the difference in power extraction can rapidly increase. This result
illustrates the importance, for the power optimisation of the system, of the effects of
interaction between devices.
4.5.3
A Row of Two OWC Devices
4.5.3.1 Analysis
1
2
Figure 4.7: Schematic Diagram of the arrangement of a row of two identical, fixed,
cylindrical OWC devices.
In this section, we are considering a row of two identical, fixed, cylindrical
OWC devices with finite wall thickness. The devices and the spacing between them
are exactly the same as in the previous section.
Due to symmetries and to the reciprocity relations (4.25), the following
equalities can be deduced
90
Moreover, the radiation problems of this system are exactly the same as in the
previous section (Section 4.4.1), it follows that the coefficients
and
,
,
are equal to those of the previous section.
Solving equation (4.23) for this problem, and taking into accounts the relations
in (4.51), it can be shown that the pressures for each of the devices can be expressed
as follow
and
The total mean capture width of the system can be obtained as follow
In the previous section, we investigated the effect of air compressibility in the
optimised total mean capture width
. We can expect the air compressibility to
91
have a similar effect in this problem. We are, in this section, focusing on the
computation of the maximum total mean capture width of the system
The expressions of
and
.
are, for this case, more complex, and finding
a closed form of the optimum parameters
,
,
and
becomes
difficult. For this reason, we considered applying a numerical multi-variable
optimisation function on
at each frequency. After testing
different optimisation methods, we found that routines which do not use gradients are
usually more stable. This comes from the same reasons stated in Gomes et al. (2012);
the different parameters of the system are computed using the FEM model and a
numerically computed gradient may mislead the right direction and can ensue in
noisy results. In this study, we applied an optimisation algorithm based on the
Nelder-Mead Simplex Method developed in Lagarias et al. (1998).
Simplex methods use the vertices of a simplex to interpolate a linear
polynomial. A new vertex is found by maximizing the linear polynomial inside a
trust region with a prescribed radius and subject to the constraints of the problem. If
the objective function value, at that vertex, is higher than at any of the others vertices
of the simplex, the vertex with the minimum objective function value is substituted
by the new one. This step is repeated until convergence is found. When no
improvement is verified, the radius of the trust region is reduced. Generally,
convergence is found when the radius of the trust region reaches a given value,
which normally defines the accuracy of the solution.
One of the key criteria for convergence in a Simplex method, as in most
iterative algorithms, is the preliminary values of the variables before iteration. We
found that convergence was fast and accurate when using, as a starting point, the
parameters
,
,
and
obtained in the hypothetical case where the
devices are not interacting with each other:
92
Due to the constrains imposed on
and
, if either of them resulted with
a negative value, the method was reapplied by setting the negative parameter
0 and the related turbine parameter
to
to
.
4.5.3.2 Results and Discussions
Figure 4.8 presents the different free surfaces as in the previous section. Of
interest, in this array, is the loss of symmetry between the two devices. Even if the
radiation problems are the same as previously, when rebuilding the free surface with
the optimum pressures
and
,
does not exhibit a symmetry between
the devices. This loss of symmetry demonstrates that the use of the maximum turbine
parameters induces different pressures inside the two devices.
From Figure 4.9 and 4.10, we can clearly see that the excitation coefficients,
the optimum turbine parameters, the optimum air compressibility parameters and the
maximum capture widths differ depending on the position of the device in the array.
The values obtained can be significantly different between the devices and those of
the SIOD. For example, the maximum values of the excitation coefficient amplitudes
give
,
6.2 whereas
shift in position is observable
, for the SIOD, and even a slight
3.3, 3.18 and 3.21 respectively.
The maximum mean capture width
exhibits a more complex behaviour
than previously, being sometimes higher and sometimes lower than for the SIOD. It
is important to remark that
and
do not represent the maximum
capture widths which could be obtained from each device independently. However, if
more power is extracted from one of the devices, it would impact on the power
extraction of the other and would entail a decrease in the maximum mean capture
width of the overall array. We can also notice that when calculating the optimum
capture width
, for any desired chamber volumes of air
would equal
only at frequencies where
and
and
,
. In
regards of Figure 4.9.b, that could possibly mean the choice of different chamber
volumes of air depending of the position of the device in the array. The choice of
different volumes of air could also be a way to increase the overall frequency power-
93
capture band-width of the system by inducing compressibility resonance at different
frequencies depending on the position of the device. It follows that even the physical
properties of the OWC device might be chosen depending on its position in the array.
a)
b)
c)
d)
Figure 4.8: Dimensionless free surface amplitudes attached to a) the radiation
problem 1,
, b) the sum of the radiation problems,
, c) the diffraction
problem,
, and d) the overall problem,
. The frequency is
= 3, and the
parameters of the turbines and air compressibility are
,
,
and
.
94
a)
b)
Figure 4.9: a) Dimensionless amplitude of the complex excitation coefficients
,
compared with
of the SIOD. b) Dimensionless parameters of
the turbine
and
,
and dimensionless parameters of air compressibility
. The -axes represents the dimensionless frequency
.
95
Figure 4.10: Dimensionless capture widths
,
,
compared with
of the SIOD. The -axes represents the dimensionless frequency .
4.5.4
Two Rows and Two Columns of OWC Devices
4.5.4.1 Analysis
1
2
3
4
Figure 4.11: Schematic Diagram of the two-rows and two-columns arrangement of
four identical, fixed, cylindrical OWC devices.
In this section, we are considering four, fixed, cylindrical OWC devices with
finite wall thickness. The devices are placed in a two-rows and two-columns
96
arrangement and numbered from 1 to 4 as presented in Figure 4.11. The dimensions
of the OWCs and the spacing between row and columns are exactly the same as in
the two previous sections.
Due to symmetries of the system and the reciprocity relations (4.25), the
following equalities can be deduced
Similarly as in Section 4.1.1, one of the purposes of this study is to find the
optimum turbine parameters in order to maximise the power extraction of the system.
Due to symmetry and considering that the devices possess the same chamber volume
of air
, we can presume that the turbine parameters, and therefore the dynamic
pressures, the volume fluxes, and power extractions for the two OWC devices in one
column should be the same,
Solving equation (4.23) for this problem, and taking into account the relations
in (4.56) and (4.57), it can be shown that the pressures for each of the devices in a
row can be expressed as follow
97
and
The total mean capture width of the system is
As in Section 4.4.2, we are only interested on the maximum total mean capture
width
of the system and the same Nelder-Mead simplex optimisation
98
algorithm was applied to
for each frequency, with the
similar iteration starting point
Once again, if either
or
resulted as a negative value, the method
was reapplied by setting the negative parameters to be 0 and the related turbine
parameter
to be
.
a)
b)
c)
d)
Figure 4.12: Dimensionless free surface amplitudes attached to a) the radiation
problem 1,
, b) the sum of the radiation problems,
, c) the diffraction
problem
, and d) the overall problem,
. The frequency is
= 3, and the
parameters of the turbines and air compressibility are
,
,
and
.
99
a)
b)
Figure 4.13: a) Dimensionless amplitude of the complex excitation coefficients
,
and
related to the SIOD. b) Dimensionless radiation
conductances and radiation susceptances
,
-axes represents the dimensionless frequency .
and
,
for the SIOD. The
100
a)
b)
Figure 4.14: a)
frequency .
,
. b)
,
. The -axes represents the dimensionless
101
a)
b)
Figure 4.15: a) Dimensionless parameters of the turbine
dimensionless parameters of air compressibility
capture widths
,
,
,
and
The -axes represents the dimensionless frequency .
and
,
and
. b) Dimensionless
related to the SIOD.
102
4.5.4.2 Results and Discussions
Similar results to the previous sections were computed. The different free
surface amplitudes are presented in Figure 4.12 and the properties of the system are
presented in Figure 4.13, 4.14 and 4.15.
Even though several symmetries are kept in this problem, the behaviour of the
system becomes considerably more complex. The inner properties of the OWC
devices present more and more discrepancies with those of the SIOD. Such
discrepancies can be seen in Figure 4.13.a for the complex excitation coefficients and
in Figure 4.13.b for the radiation conductances and radiation susceptances. From
Figure 4.14.a and 4.14.b, the values of the radiation conductances and the radiation
susceptances, induced by the other pressures, exhibit more complex and distinct
behaviours. It becomes clear that effects of other devices’ radiated waves are
strongly dependent on the relative position of the influenced and the influencing
device. From Figure 4.15.a and 4.15.b we can see that the optimised parameters of
the turbines, the optimum chamber volumes of air and the resulted maximum capture
widths become even more dissimilar depending on the position of the device in the
array. This is especially evident around the natural resonance frequency of the
system. Moreover the capture widths
and
display greater differences, as
seen in Figure 4.15.b.
4.6
Conclusion
In this chapter, the theory of the interaction between oscillating systems,
developed by Falnes & McIver (1985), is for the first time applied and extended to
an explicit study of the dynamic and energetic performance of a finite array of fixed
OWC devices with Wells type turbines as power take-off. This extended method
considers the coupling between pressures and volume fluxes, the interactions
between the devices, and the effects of air compressibility. Once again, this method
is very efficient as only a set of hydrodynamic coefficients are needed in order to
study the dynamic and energetic behaviour of the overall system.
Following this method, the FEM model was applied to the study of three
different array configurations: a column of two identical cylindrical OWC devices, a
row of two identical cylindrical OWC devices, and two rows and two columns of
identical cylindrical OWC devices. As in Chapter 2, an array of devices can behave
103
quite differently to a single isolated OWC device. In this chapter, it has been
demonstrated that the inner properties of the OWC devices and the radiation
influences between devices are strongly dependent on the position of the device in
the array. It is also revealed that the dynamic and energetic performance of the
system becomes more complex and distinct from the SIOD as the number of devices
in the array increases.
As in Chapter 2, more hydrodynamic power can be extracted from the array, at
some frequencies, than would be extracted from the sum of the same number of
isolated devices. Optimisation of the overall power extraction of the system is
performed. A closed form of the optimum parameters is derived in the first problem
studied. However, due to the significant increase in complexity of the overall mean
hydrodynamic power expression, an optimisation is then performed, in the two other
problems, through the use of the Nelder-Mead Simplex Method applied to the total
mean capture width of the system. It is shown that taking into consideration the
coupling between devices increases the overall power extraction of the system. The
results also suggest that the position of the device in the array should be taken into
account when determining device parameters, such as the chamber volume of air, so
as to increase the maximum power extraction of the system or the overall frequency
power-capture band-width.
In practice farms of OWC devices are more likely be deployed than widely
separated individual OWCs so as to harness maximum available power and to
facilitate installation and electrical power transmission. The results from Chapter 2
and this chapter, demonstrate that such systems will need to be specifically designed.
Spacing between devices, physical properties of each of the devices in the array and
their turbine characteristic can be specifically devised in order to maximise the
power extraction of the system. The method developed in this chapter and its
application through numerical models, such as the present FEM model, is believed to
have the potential to efficiently assist in OWC array design.
104
5 HYDRODYNAMIC AND ENERGETIC PROPERTIES OF A MOORED
HEAVING OSCILLATING WATER COLUMN WAVE ENERGY
CONVERTER
5.1
Introduction
Oscillating Water Column (OWC) devices can be installed on the coast in
order to reduce the overall installation cost. As presented in Chapters 2 and 3, these
systems are often fixed. The power extraction, for a given geometry, is then
completely dependent on the incident wave properties as well as the properties of the
turbine. Devices can also be installed further away from the coast for greater wave
power availability. In the latter situation OWC devices are mostly floating structures.
The motion of the structure then becomes an important factor in the power extraction
of the system. The volume flux going through the turbines and, therefore, the power
extraction depend on the relative motion between the water and the body. Mooring
properties and air pressure inside the chamber can also influence the motion of the
device.
General numerical modelling of fully floating OWC devices has been reported
by Sykes et al. (2009) and Hong Hong et al. (2004). More recently, in-depth research
of the hydrodynamic properties of a heaving OWC spar buoy was performed by
Falcão et al. (2012) and Gomes et al. (2012). In this studies, the coupling between
the different degrees of freedom and the oscillating pressure follows the interaction
theory developed by Falnes & McIver (1985) in order to investigate the energetic
optimisation of the system. However, in order to derive the various hydrodynamic
coefficients required, the upper part of the water column was modelled as a heaving
piston. As discussed in Chapter 1, the reason for this is likely to be around the
difficulties that confront those modelling enclosed chambers using a BEM-based
model with a source distribution approach such as WAMIT.
Such difficulties are avoided when using a FEM based model. In this chapter,
the interaction theory is applied to a heaving OWC device. Direct coupling between
the motion of the device, the pressure inside the chamber, the volume fluxes and the
forces are considered.
Following this method, the 3D FEM model was then applied to a heaving
cylindrical OWC device with finite wall thickness in order to the study the dynamic
and energetic behaviour of the system in waves. The study especially focuses on the
105
effect of air compressibility, the optimisation of the turbine parameters and the
effects of the mooring restoring force coefficient.
5.2
5.2.1
Formulation
General Boundary Conditions
Figure 5.1: Schematic diagram of a single floating isolated OWC device
The OWC device considered in the present study is a truncated cylinder with a
finite wall thickness. The cylinder is surface piercing and operates in constant water
depth . The inner radius is , the outer radius , creating the cross-sectional surface
. The draft of the cylinder is
. A Cartesian coordinate system
corresponding cylindrical coordinates
with its
are situated with the origin coincident
with the centre of the cylinder at the mean sea water level, the -direction pointing
vertically upwards as illustrated in Figure 5.1. A monochromatic plane wave of
amplitude
and frequency
propagates from
. Linear water-wave theory
is assumed and with the assumptions of irrotational and inviscid flow, a velocity
potential
exists that satisfies the Laplace equation:
Under these assumptions
value
as
can be expressed using its corresponding complex
106
In the same way as in the previous chapters, the computational domain is
separated into two regions; an outer region between a radius of
complex velocity potential
within the radius
and
with a
, and an inner region, containing the OWC device,
with a complex velocity potential
. Both potentials satisfy the
Laplace equation
and
The velocity potential
velocity potential
can be decomposed into the sum of the incident wave
and the velocity potential
induced by the scattering of the
wave by the device as
where
In this expression,
equal to
is the gravitational constant and
, the parameter
the dispersion relation
is the wave number,
being the wavelength. The wave number
satisfies
107
A uniform pressure
is applied in the air chamber and the heave motion as
one degree of freedom of the device is also considered. Both the displacement
from the equilibrium of the device in the
direction, and the pressure
are believed
to oscillate around their equilibrium value at the same frequency ω of the incoming
wave. Under these assumptions,
corresponding complex values
, and
, and
can be expressed using their
as
and
The general boundary conditions for this problem can be expressed as follows:

In the outer domain
:
On the sea floor at
On the surface, at
And the Sommerfeld radiation condition on

Between the two regions at
:
when
tends to infinity
108

In the inner region
:
On the device walls at
and
On the cross-sectional surface
,
of the device at
,
On the surface outside the OWC chamber, at
On the surface
where
5.2.2
created by the water at rest inside the chamber,
is the density of the water.
Heave Motion and Dynamic Pressure
As previously, the dynamic pressure
total volume flux
is considered to be proportional to the
going through the turbine. By taking into account air
compressibility, as in Sarmento & Falcão (1985), the relationship between the
dynamic pressure
be expressed as
and volume flux
, inside the chamber of the OWC device can
109
where
and
.
is the air density at atmospheric pressure,
the air,
is the rotational speed of turbine blades,
rotor,
is the velocity of sound in
is the outer diameter of turbine
is the air density at atmospheric pressure and
the chamber at rest.
the air volume inside
is an empirical positive coefficient which depends on the
design, the number and set-up of turbines.
The total volume flux in the chamber is, for this problem, dependent on the
relative volume flux between the volume flux induced by the oscillation of the free
surface inside the chamber and the heave motion of the device.
can be expressed
as follow
where
and
is the volume flux created by the oscillation of the free surface,
is the volume flux created by the heaving motion of the device,
The expression (5.17) becomes
Disregarding viscosity effects, the displacement of the device is considered to
follow the equation of motion
110
where
is the total mass of the device.
corresponds to the hydrodynamic force
exerted by the hydrodynamic pressure on
is the forced induced by the pressure
on the device at the top of the air
chamber and is approximated to
is the force induced by the mooring system on the device. The magnitude
and influence of such a force is strongly dependent on the type of mooring used.
However, the spring effect is usually one of the most prominent influences of the
moorings. In this study, involving small wave and structure amplitudes, we therefore
only consider the mooring spring effect and other effects such as linear damping or
inertia effects are disregarded.
where
is therefore idealised as
is the moorings restoring force coefficient and is considered positive and
constant.
Under hydrostatic equilibrium, the following relationship is obtained
Introducing complex values for the different variables, the equation of motion
(5.23) becomes
111
where
From (5.22) and (5.28), it is apparent that the heave motion and the dynamic
pressure inside the chamber are, in this problem, fully coupled. By rearranging (5.22)
and (5.28), the displacement
and the dynamic pressure
function of the hydrodynamic force
and the volume flux
can be expressed in
, giving
and
The boundary conditions (5.14) and (5.16) become
and
For given turbine and mooring parameters, introducing expressions (5.29) and
(5.17) for
and
into the boundary conditions (5.32), (5.33) gives the overall
boundary conditions in terms of the velocity potentials
can theoretically be solved.
and
only and the system
112
Special cases can be deduced from these equations. Considering the turbine
coefficient
to tend to infinity gives the boundary conditions for the floating OWC
device without pressure inside the chamber,
and
whereas, considering the restoring force coefficient
to tend to infinity gives the
boundary conditions for the fixed OWC device with pressure inside the chamber
5.3
Analysis
Under linear water wave theory, it is possible to separate the velocity potentials
into
is the diffracted wave potential induced by the interaction between the
incident wave and the OWC device in the case where no pressure is present inside
the device chamber and the device is fixed at its equilibrium state.
satisfies the
boundary conditions developed in Section 5.2.1 by considering the pressure
the displacement
to be zero.
and
is the pressure radiated wave potential induced
113
by the forcing on the water of the oscillating pressure
inside the chamber.
satisfies the boundary conditions developed in Section 5.2.1 by considering the
incident velocity potential
and the displacement
to be zero. Finally,
is the
heave motion radiated wave potential induced by the effect of the heave motion of
the device on the water.
satisfies the boundary conditions developed in Section
5.2.1 by considering the incident velocity potential
The volume flux
and the pressure
to be zero.
induced by the oscillation of the water inside the chamber
and the hydrodynamic force
on the surface
can be separated into
and
Following the interaction theory between oscillating systems, presented by
Falnes & McIver (1985), we express each of the contributions as
where the coefficients
,
,
,
,
and
is the volume flux excitation coefficient;
displacement excitation coefficient and
are complex numbers.
is the radiation admittance;
is the
could be directly related to the added
mass and radiation damping due to the heave motion of the OWC body.
are the hydrodynamic coupling coefficients.
and
reflects the contribution of the
oscillating pressure to the total hydrodynamic force on the OWC body and
reflects the contribution of the heave motion to the total water volume flux inside the
chamber. Moreover, Falnes & McIver (1985) demonstrated that
114
The coefficients
,
,
,
,
and
only dependent of the incident wave frequency
are, for a given geometry,
.
and
would also be
dependent on the direction of the incident wave propagation and the -axes for a
non-axisymmetric device, if considered.
The expressions (5.22) and (5.28) become
and
The pressure
and the displacement
can then be expressed by use of the
different coefficients
and
The mean hydrodynamic power extracted by the system can be expressed as
115
where
denotes the complex conjugate of the pressure
. From (5.45), it gives
The power capture width is given by
Cg being the group velocity of the incident wave.
The different
,
,
,
and
are, in this chapter, the key
parameters of the study. They can be computed for a given geometry and incident
wave frequency
. Dynamic pressure, volume flux, motion of the device, mean
hydrodynamic power extraction, and capture width can then be directly obtained for
any desired mooring parameter
parameters
, volume of air inside the chamber
and turbine
.
It is, in practice, difficult to change the mooring properties and the volume
inside the air chamber, however, after choosing appropriate moorings and volume, it
is then possible to optimise the turbine parameters in order to achieve maximum
hydrodynamic power extraction. If we introduce the complex parameter
real parameters
and
and
such as
, and the
116
it becomes apparent that relation (5.51) becomes identical to relation (3.30) for a
fixed device
and
can be seen as the overall radiation conductance and the overall
radiation susceptance of the system.
Optimum power extraction and capture width can then be obtained using the
turbine parameters
giving
and
In this study, as in the previous chapters, we will also be interested in the
overall maximum mean hydrodynamic power available to the system when a typical
Wells type turbine is considered for power take-off. Depending on the sign of
parameters for maximum hydrodynamic power are
the
117
when
is negative or zero, leading to
and
or
when
and
is positive, giving
118
5.4
The Finite Element Model
Figure 5.2: Example of a mesh used around the OWC device
The FEM used in this chapter is identical to the one described in Chapter 3 and
the second-order cylindrical damper, as prescribed by Zienkiewicz et al. (2005), was
applied at the spatial limit of the model, at
, for the diffraction problem and
the radiation problems, as will be described subsequently,
where
is the tangential coordinate, at
and is equal to
In order to derive the coefficients
,
,
,
.
and
, necessary to
the study, the problem was separated into the diffraction problem, the radiation
problem induced by the oscillation of the pressure inside the chamber and the
radiation problem related to the heaving of the device, as described in Section 5.3 for
the velocity potentials
,
and
.
119
Through the diffraction problem, the complex excitation coefficients
and
could be computed by deriving the volume flux inside the chamber and the
hydrodynamic force on the surface
and
,
,
and
being equal to 0 in this problem.
The model was next applied to the radiation problem related to the oscillating
pressure inside the chamber. The radiation admittance
coupling coefficient
, and the hydrodynamic
could then be computed
and
,
,
and
being 0 in this problem.
The model was finally applied to the radiation problem related to the heaving
motion of the device and the added mass and radiation damping coefficient
could be obtained as follow
120
and
being 0 in this problem.
As previously, once the pressure
and the heaving motion
derived from equations (5.45) and (5.46), using the desired parameters
have been
,
and
, the radiation solutions can then be reapplied with the appropriate pressure and
displacement. The total velocity potential
and other hydrodynamic properties (free
surface, pressure, velocity, etc.), all over the fluid, can be computed.
5.5
5.5.1
Results and Discussion
Non-Dimensional Parameters
Prior to presenting and discussing the different results obtained from the
model, we shall introduce the non-dimensional form of the properties of interest.
The non-dimensional free surfaces are defined as
where ,
,
and
represent the total free surface, the diffracted wave free
surface, the pressure radiated wave free surface and the displacement radiated wave
free surface, respectively.
The dimensionless turbine parameter
overall radiation conductance
The different coefficients
as follow
, air compressibility parameter
and overall radiation susceptance
,
,
,
and
,
are chosen as
are undimensionalised
121
The non-dimensional volume fluxes are defined by
And finally, the dimensionless capture width is symbolised by
5.5.2
Properties and Air Compressibility
In the following, the dimensionless parameters defining the system were
selected so as to be comparable to the previous chapters: inner radius
outer radius
= 0.25 and the draft of the cylinder
this section is considered to be freely floating (
= 0.2,
= 0.2. The OWC device, in
0). The model was applied to
the diffraction problem and the different radiation problems for a set of frequencies
comprised between
and
.
As discussed previously, after having computed the different hydrodynamic
coefficients
,
,
,
and
and having derived the pressure
displacement
for any desired chamber volume of air
and the
and turbine property
,
the model can be reapplied using the appropriate parameters. The results of the
different problems can be added to obtain the total velocity potential over the entire
numerical domain. Hydrodynamic properties such as the free surface, pressures,
velocities, etc., can also be computed.
122
a)
b)
c)
d)
Figure 5.3: Dimensionless free surface amplitudes a)
, b)
, c)
and
d)
. The frequency is
= 3, and the parameters of the turbine, air
compressibility and the mooring restoring force coefficient are
,
and
0N.m-1.
The different dimensionless free surface amplitudes around and inside the
OWC device chamber are presented in Figure 5.3. Figure 5.3.a, 5.3.b, 5.3.c and 5.3.d
represent the diffracted-wave free surface amplitude
wave free surface amplitude
amplitude
, the pressure radiated-
, the heave motion radiated-wave free surface
and the reconstituted total free surface amplitude
The frequency of the wave is
compressibility are
and
by the heaving of the device
the OWC device than
, respectively.
, and the parameters of the turbine and air
. It is noticeable that the radiated wave induced
generates significantly higher amplitudes around
. The pressure radiated is contained by the walls of the
chamber where the heave radiated wave is not. As discussed previously for the fixed
123
case,
is close to the natural resonance frequency of the water column. Large
amplitudes are therefore noticeable inside the OWC chamber for each of the different
wave problems. The pressure radiation problem is identical to the fixed case.
However, compared to Figure 3.3.b, slightly different values are noticeable. This
difference in results is expected as the optimum pressure is now also dependent on
the heaving motion of the device. We can also observe, in Figure 3.3.c, that the total
free surface amplitude maxima and minima and the gradient in amplitudes are more
pronounced around the OWC device for the floating case than for the fixed case. The
total radiated waves field is stronger when added the heave motion and the
interaction with the scattered wave field is therefore more pronounced.
The real and imaginary parts of the hydrodynamic coefficients
,
and the overall radiation conductance
,
,
,
and radiation susceptance
are
presented in Figure 5.4, 5.5 and 5.6. It is interesting to observe that, apart for
and
, each of the coefficient components, even those related to the heaving motion,
and
, displays a particular behaviour around the natural resonance frequency of
the water column. These behaviours are characterised by a maximum, a minimum or
an inflexion point at
3.21.
It is clear from Figure 5.6.b, that
coefficients. As in the fixed case,
whereas
and
behave differently to the other
is typically positive over all frequencies
changes sign. However, the change of sign in
and the maximum in
represent a divergence in their value. This divergence is situated at the local
frequency
4.54. This frequency represents the zero of the term
from the equation (5.51). When no pressure is present inside the OWC
device, it can be shown that the expression (5.46) of the displacement
The frequency
motion of the device.
becomes
4.54 is the natural resonance frequency of the heaving
124
a)
b)
Figure 5.4: Real and imaginary parts of the dimensionless hydrodynamic coefficients
a)
and b)
, against
.
125
a)
b)
Figure 5.5: Real and imaginary parts of the dimensionless hydrodynamic coefficients
a)
and b)
against
.
126
a)
b)
Figure 5.6: Real and imaginary parts of the dimensionless hydrodynamic coefficients
a)
and b) and against
.
127
The total volume flux
, the volume flux induced by the oscillation of the
water inside the chamber
device
and the volume flux induced by the heaving of the
are presented in Figure 5.7a for the case where no pressure is present
inside the chamber. Both
and
exhibit maxima at the natural frequency of
the water column and at the natural frequency of the heaving of the device. This
result mirrors the mutual influence that the oscillation of the water has on the
heaving of the device and vice-versa. It is noteworthy that the heaving of the device
passes through a zero value at
3.73. This value coincides with the zero values
in both the imaginary part and real part of the displacement excitation coefficient
in Figure 5.7.b. Before this frequency, the sum of
destructive on the total volume flux amplitude
either
or
. Both
and
and
, meaning that
is mostly
is lower than
tend to 1 for low frequencies whereas
tends to zero. This shows that for long wave, the water inside the chamber and the
heave motion of the device tend to oscillate at the same amplitude and phase. After
3.73,
than
and
and
show more constructive behaviour, meaning that
is higher
. For each of the components, a local divergence, at the heaving
resonance frequency,
4.54, is perceptible. The resonance peak is, however,
particularly narrow-banded. It is evident that such divergence in amplitude, at the
natural resonance frequency of the heaving motion, would not exist in reality
because, for high motions of the device, viscosity will generate non-negligible
damping. Then, for high frequencies, each of the volume fluxes tends to zero.
In Figure 5.7.b and 5.8, effects of the air compressibility on the total volume
flux and mean power extraction is investigated. Optimum parameters of the turbine
are used for different air chamber volumes of air
results are also compared with those using the parameter
0, 5, 10, 50
and
. The
. As for
the fixed case it is informative to consider the sign of the overall radiation
susceptance
. Interestingly, for the freely floating case,
displays negative values
only after the natural frequency of the heaving motion. Prior to this frequency, as the
volume of air increases, the volume fluxes tend to the no-pressure case. In this low
range frequency, maximum hydrodynamic power extraction is found for
0m3
and decreases as the volume of air increases. It is noteworthy that local maxima are
128
found at the natural resonance frequency of the water column but are significantly
smaller than for the fixed case.
The frequency
4.54 is something of an oddity, since neither
,
nor
possess a finite value at this frequency. However, each of the results, whatever
the volume of air, behaves in the same way as if the relation
The same finite maxima in the volume flux amplitude,
capture width,
was fulfilled.
12.4, and in the mean
0.83, are found, at this frequency, for all the air chamber
volumes considered.
0.83 also represents the overall maximum in the
capture width. In application, it will certainly be difficult to set up the proper turbine
properties when the frequency of the wave is close to
particularly narrow-banded and the values of
4.54. The peak is
are very high.
In the higher frequency domain, each time the expression
is fulfilled,
a maximum in the volume flux and in the capture width is exhibited. In contrast to
the fixed case,
will be satisfied once for any volume of air considered. The
exception is the case where
0; the solution can be conceptualised as having
been pushed to an infinite frequency. The higher the volume of air, the closer to
4.54 the peak occurs, and the smaller the volume of air the further away the peak
from 4.54.
In the low frequency region, when
is positive, the parameters
are by definition the same as when
0. As
increases away from the
heaving resonance frequency, in the negative region of
related to the parameters
and
and
, the volume flux
,
, gradually increases whereas
appears to linearly but slowly decline. As previously seen for the fixed case,
passes through the different
maxima as
increases.
As in Chapter 3, the air compressibility can significantly increase the capture
band width of the power extraction. This is especially noticeable for
10
related to
with three maxima at each of the water column, heaving motion and
air compressibility resonance frequencies. However, the values within this bandwidth are mostly low with an average of around 0.15. The large amplitude peaks at
and
are very narrow and the value of the capture width at
3.21 is significantly smaller than for the fixed case.
129
a)
b)
Figure 5.7: a) Non-dimensional volume flux amplitudes
,
and
when
0. b) Dimensionless amplitude of the total volume flux
for different cases,
for
0, using the turbine parameter
for different chamber volumes of air,
0, 5, 10, 50
and using the parameters
and
. The -axis
represents the dimensionless frequency
and
0.
130
Figure 5.8: Dimensionless optimum capture width
for different chamber
volumes of air
0, 5, 10, 50
and dimensionless overall maximum capture
width
. The -axis represents the dimensionless frequency
and
0.
5.5.3
Effects of the Mooring Restoring Force Coefficient
The restoring force coefficient
, representing the effect of the mooring
system, is present in all the equations (5.45) to (5.51) related to the behaviour of the
device. From the previous section, it is obvious that the performance of the freely
floating OWC device is significantly different to that of the fixed OWC device.
However, from the different expressions it follows that as
increases, the dynamic
of the system should tend to that of the fixed case. In order to investigate the effects
of the mooring restoring force coefficient, severall values of
0, 0.1, 0.5, 1, 5
suseptance
. The overall radiation conductance
, the total volume flux
presented, for the different values of
were tested,
, the overall radiation
and finally the mean capture width
are
, in Figure 5.9.a, 5.9.b, 5.10.a and 5.10.b,
respectively. As shown in the previous section, air compressibility has a significant
influence on the behaviour of the device. In order to remove the dependency on
but still be able to visualise the effect the air compressibility, in this study the present
131
author considered the case with
with
0 and the case with
. By comparing the two cases and by considering the values of
and
, it is possible
to understand these influences.
As expected, for large mooring spring coefficient
, the results become
similar to the fixed case. The local maxima for the radiation conductance
volume fluxes
, and the capture widths
and
the natural resonance frequency of the water column.
slightly below
, the
typically increase at
exhibits a local maximum
3.21 and a local minimum slightly above. As
increases, the
natural resonance frequency of the heaving motion is shifted toward higher
frequencies. This is understandable as the zero of component
, in equation (5.74), moves to higher frequencies. In between the lowest and
highest values of
used, a very interesting behaviour in the power extraction of the
device appears. A new significant capture width maximum emerges. This maximum
becomes noticeable for
observable for
frequency
related to
but is most clearly
related to
. This maximum is located at the
4.2 with a value
0.67. Remarkably, this maximum is not
visible in the total volume flux but it can be seen that it coincides with a zero value
of
. As we noticed previously,
was exhibiting a local maximum and a local
minimum around the natural resonance frequency of the water column. Starting with
,
displays negative values around its local minimum. The first zero
happens near the water column natural resonance frequency tending to the fixed case
as
increases. Differently to the fixed case,
becoming positive once more. As
passes through a new zero before
increases, this new zero is shifted toward
higher frequencies and so is the new maximum in
. The values of
being
negative between the two zeros discussed, higher values of the capture width can
potentially be obtained due to air compressibility effect and are illustrated by
. The value of the capture width would still be dependent on the choice of the
chamber volume of air.
132
a)
b)
Figure 5.9: a) Dimensionless overall radiation conductance
and b) dimensionless
overall radiation susceptances
for different restoring force coefficients
0,
0.1, 0.5, 1, 5
compared with the fixed device results. The -axis represents the
dimensionless frequency
133
a)
b)
Figure 5.10: a) Dimensionless total volume flux amplitudes
for
and
0 and for
and
and b) dimensionless optimum capture width
for
0 and
for different restoring force coefficients
0, 0.1, 0.5, 1,
5
compared with the fixed device results. The -axis represents the
dimensionless frequency .
134
This finding could have significant practical consequences. For the case where
, the energy band-width is much wider than for the fixed case and has a
significantly higher power extraction rate than for the freely floating case. This
shows how important the influence of the mooring properties can be on the dynamic
and energetic behaviour of a floating device. Moreover, these influences can have a
beneficial effect on the hydrodynamic power extraction rate.
5.6
Conclusion
The method of interaction between oscillating systems developed by Falnes &
McIver (1985) has been applied specifically to a heaving OWC device and extended
to take into account the relationship between the volume flux and the pressure. Direct
coupling between the motion of the device, the pressure inside the chamber, the
volume fluxes and the forces were considered. Moreover, air compressibility and the
spring effect of the moorings system were taken into account. As previously, a study
of the dynamic and energetic behaviour of the OWC device was performed through a
set of frequency dependent hydrodynamic coefficients,
,
,
,
and
. Pressures, volume fluxes, heave motion and power extraction were then directly
computed for any desired parameter of the turbine
, chamber volume of air
,
and mooring properties. The closed form of the optimum parameter of the turbine
, for maximum energy extraction, could also be derived by introducing the
overall radiation conductance
and radiation susceptance
.
The FEM model was then applied to a cylindrical OWC device. In order to
derive the hydrodynamic coefficients, the model was applied to the diffraction
problem, the pressure radiation problem and the heaving radiation problem once for
each frequency. In the first part of the study, the device was considered to be freely
floating,
0, and the effect of air compressibility was investigated. It was found
that air compressibility can increase the power capture band width of the system.
This band-width can be significantly broader than for the fixed case but the values of
the capture width were, on average, considerably lower. The exceptions were the two
narrow peaks observed at the heaving motion and air compressibility resonance
frequencies.
135
In the second part of the study, the influence of the mooring system on the
performance of the device was investigated by varying the restoring forced
coefficient
. The most significant outcome was the appearance of a new
maximum in the power capture width. This maximum could considerably widen the
power capture width compared to the fixed case and could induce significantly
higher power extraction rate compared to a freely floating device. These results
demonstrate the non-negligible influences that the mooring system can have on the
dynamic and energetic behaviour of a floating OWC device. The mooring system
therefore becomes an important parameter of the overall system performance and can
be designed so as to improve power extraction.
136
6 POWER EXTRACTION OF A FIXED OSCILLATING WATER COLUMN
DEVICE UNDER WEAKLY NONLINEAR WAVES
6.1
Introduction
In the previous chapters, various problems related to the behaviour of OWC
devices in waves were examined using linear water wave theory. The validity of
linear theory is dependent on the assumption that wave amplitudes are small in
comparison to their wavelength and to the water depth. However, studies based on
linear water wave theory have their limitations, especially when an OWC device is
placed near shore where the water depth is shallow enough that some nonlinear
effects must be taken into consideration. Moreover, OWC devices are characterised
by resonance and significant amplitudes in the oscillations of the water column
inside the chamber can appear, even for relatively small incident wave amplitudes.
The second-order wave-body interaction has seen a widespread interest since
the work from Lighthill (1979) and Molin (1979). By using the Haskind relations, it
was shown that an accurate formulation of the second-order wave loads could be
derived without directly solving the full boundary-value problem. Following this
approach, a large number of studies emerged. Eatock Taylor & Hung (1987) reexamined the theory of Lighthill and Molin and derived definitive results for the
cylindrical cylinder. Abul-Azm & Williams (1988), (1989a), (1989b) investigated
second-order diffraction loads on a truncated cylinder, arrays of vertical circular
cylinders and arrays of semi-immersed circular cylinders. Williams et al. (1990) also
studied arrays of vertical circular cylinders by comparing complete and approximate
solutions. This method is quite efficient in the estimation of the forces on the body
but it does not give any information on the free surface elevation or other
hydrodynamic properties in the surrounding environment.
In contrast to linear water wave theory, fully analytical solutions cannot be
obtained for the second-order theory due to the inhomogeneous free surface
boundary condition and the lack of a proper radiation condition at infinity. Semianalytical methods were however developed. Chau (1989), Kriebel (1990), (1992)
and Chau & Eatock Taylor (1992) focused on the full cylinder problem. The
truncated cylinder, arrays of cylinders, a bottom-mounted compound cylinder and an
arrangement of two circular cylinders were also analysed by Huang & Eatock Taylor
137
(1996), Malenica et al. (1999) and Mavrakos & Chatjigeorgiou (2006), (2009),
respectively.
Second-order numerical methods and three dimensional models were also
applied. Kim & Yue (1989), (1990) developed a method for axisymmetric bodies in
monochromatic and bi-chromatic incident waves. Loads on bodies of arbitrary shape
were studied by Lee et al. (1991) using the BEM model WAMIT. Clark et al. (1991)
applied a FEM model to a cylindrical and elliptic cylinder. Eatock Taylor & Chau
(1992) discussed the application of the BEM to linear and second-order diffraction
problems and derived the second-order free surface profile in the vicinity of a tension
leg platform as an application example. However, to the knowledge of the present
author, no second-order models have yet been applied to an OWC type problem with
dynamic pressure inside the chamber.
In the present study, the 3D finite element model was extended to employ
Stokes’ wave theory up to second order and was applied to a fixed cylindrical OWC
device. As in previous chapters, the model included the effect of pressurisation by
implementing an oscillating pressure inside the chamber. The properties of the
turbine were also taken into account through the use of a pneumatic damping
coefficient specifying the relationship between the pressure and the total water
volume flux. Second-order corrections in terms of the overall volume flux inside the
chamber and mean power output are presented here and compared with the results
obtained from linear water wave theory.
6.2
6.2.1
Formulation
Stokes’ Wave Expansion
The OWC type device considered in the present study is a truncated cylinder
with a finite wall thickness. The surface-piercing cylinder is assumed to be
suspended but fixed in constant water depth
radius is
as illustrated in Figure 6.1. The inner
and the outer radius is , creating the cross-sectional surface Sc. The draft
of the cylinder is
. A Cartesian coordinate system
cylindrical coordinates
with its corresponding
is situated with the origin coincident with the centre
of the cylinder at the mean sea water level and the -direction pointing vertically
upwards. A monochromatic plane wave of amplitude
and frequency
propagates
138
from
. By assuming that the water is inviscid and the flow is irrotational, a
velocity potential
exists and satisfies the Laplace equation
Figure 6.1: Schematic diagram of the OWC device
The kinematic and dynamic boundary conditions at the free surface
can be
expressed as
and
where g is the gravitational constant.
Following the method of perturbation expansion, we apply a Taylor series to
the boundary conditions (6.2) and (6.3) in order to obtain the conditions on the still
water surface (
),
139
The results of the Taylor series are made traceable by considering a
perturbation series on both
where k is the wave number and
and
with respect to the wave-stiffness
,
the wave amplitude (Molin (1979)),
At first and second order the kinematic and dynamic boundary therefore
become
Moreover, the solution is considered to be a set of time-harmonics in regard to
the incident wave frequency
and
added to non-periodic components.
,
,
can then be expressed in a complex value form in order to separate
the time dependency of the boundary value problem
140
The non-periodic components
and
are used to eliminate the constant
in the Bernoulli equation. We are however only interested in the periodic
components and
and
will not be directly presented in this study.
By introducing the complex values in the expressions (6.8) to (6.11) and
rearranging these conditions, the free surface components can be removed from the
boundary conditions, giving
and
6.2.2
Boundary Conditions
We now consider the computational domain to be separated into two regions:
an outer region between a radius of
and an inner region within the radius
velocity potentials
and
wave velocity potentials
and
with velocity potentials
with velocity potentials
and
and
;
. The
can be decomposed into the sum of the incident
and
and the velocity potentials
induced by the scattering of the wave by the device:
and
141
with
In these expressions,
is the wave number equal to 2π/ ,
being the
wavelength. The wave number, , satisfies the dispersion relation
The general boundary conditions for this problem can be expressed as follows:

In the outer domain
On the sea floor at
On the surface, at
:
142
The radiation condition on
when
and a valid radiation condition for
tends to infinity is then
needs to be applied on
but this issue will
be discussed later in relation to the finite element model section 6.4.

Between the two regions at

In the inner region
:
On the sea floor at
At any point on the device walls
:
143
where
is the derivative in the direction of the unit vector
normal to the
surface of the wall and pointing outward from the fluid.
On the surface, at
and
To account for the effect of the turbine, a pressure
is considered uniform
inside the chamber. The dynamic boundary condition (6.3) becomes
where
is the density of the water.
The pressure
is also considered to be constituted of a set of time-harmonics
in respect to the incident wave frequency
. By reapplying the perturbation
expansion method and introducing the complex pressure,
and
it can be shown that the boundary conditions at the surface, at
become
,
and
144
6.2.3
Expression for the Pressure
As in previous chapters, we consider the chamber pressure
proportional to the total volume flux
to be linearly
through the chamber, approximating the
effect of a typical Wells type turbine as the power take-off system,
However, as a first step in the study of second-order non-linear effects, we
overlooked the effect of air compressibility inside the chamber. The pneumatic
damping
was considered to be real and positive.
For a fixed OWC device, the volume flux is equal to the volume flux
induced by the variation of the free surface elevation
separating
inside the chamber. By
into its first- and second-order complex terms,
we obtain
and
(6.10) and (6.11)
are derived from the surface kinematic boundary conditions
145
The boundary conditions (6.35) and (6.36) can then be expressed as a function
of
and
only
For a given geometry, incident wave frequency
, and pneumatic damping ,
the system can now theoretically be solved, providing that a valid radiation condition
is applied to the second-order problem.
6.3
Analysis
The first-order potential can be analysed in the same way as presented in
Chapter 3 for linear water wave theory. The fact that the pneumatic damping
coefficient
is, here, considered positive can simply be seen as the special case of
Chapter 3 where
as
and
0. The first-order pressure can then be expressed
146
where
,
and
are the first-order complex excitation coefficient, the first-
order radiation conductance and the first-order radiation susceptance, respectively.
The perturbation method has in effect to linearise, at each order, the different
boundary conditions. It is noticeable from the results in the previous sections, that the
non-linear quadratic terms left in the second-order equations, are comprised of firstorder components only and these equations are fully linear in second-order
components
and
. In this analysis, we separate the second-order velocity
potential into a diffracted wave potential
forced-wave potential
, a radiated wave potential
and a
,
The diffracted wave potential
is similar to the diffracted wave potential of
the linear wave problem and represents the effects of the interaction between the
incident wave potential
numerical domains,
and the OWC device. After separation of the two
satisfies the general boundary conditions presented in
Section 6.2.2, except that the free surface boundary conditions (6.25), (6.32) and
(6.36) are replaced by homogeneous boundary conditions (the terms in the right hand
side of the equations are all set to zero).
The radiated wave potential
is identical to the radiated wave potential of
the linear wave problem and accounts for the effects of the forcing on the water of
the oscillating pressure
.
satisfies the general boundary conditions presented
in Section 6.2.2 by simply considering the potential
to be zero. The free surface
boundary conditions (6.25), (6.32) are also replaced by homogeneous boundary
conditions and the first-order terms in the boundary condition (6.36) are removed,
leaving the pressure term only.
Finally, the forced-wave
is a specificity of the second-order theory and
was best described by Kriebel (1990).
represents the effects of the first-order
forcing terms in the inhomogeneous free surface equation. These forcings can be
conceptualised as a number of continuous local pressures oscillating on the free
147
surface and creating an infinite number of radiated waves interacting with each other
and the OWC device.
satisfies the general boundary conditions presented in
Section 6.2.2, by considering the potential
The waves associated with the potentials
and the pressure
and
to be zero.
can be defined as free
waves, meaning that they can propagate freely on the free surface. It follows that
both the diffracted wave and the radiated wave satisfy the dispersion relation
where
is the wave number attached to free waves oscillating at frequency 2 .
As described by Kriebel (1990), the forced wave can be conceptualised as
comprised of an infinite amount of local forcing induced by the inhomogeneous free
surface condition, as well as an infinite number of free-waves, effect of these forcing.
The local forcing part of the forced wave is locked in phase with the first-order
potential and it follows that the forced wave do not satisfy the second-order
dispersion relation (6.46).
The volume flux
where
and
inside the chamber can be separated into
148
As with the first-order diffraction problem,
can be considered proportional
to the square of the incident wave amplitude
where
is the second-order complex excitation coefficient.
Similarly the first-order radiation problem,
can be considered proportional
to the second-order dynamic pressure
where
and
are the second-order radiation conductance and radiation
susceptance, respectively. Moreover, due to the similarity between the second-order
and first-order radiation problems, we can deduce the following relationships:
Finally, as a result of the quadratic first-order forcing,
can be considered,
in this problem, to be proportional to the square of the incident wave amplitude
where
is the forced-wave excitation coefficient.
The coefficients
,
and
a given geometry. However, since
coefficient
, are only dependent on the frequency
is dependent on first-order properties, the
is therefore a function of the frequency
coefficient .
The pressure
for
can then be expressed as
and the damping
149
The mean hydrodynamic power extracted by the system can be expressed as
follows:
is the first-order wave period and is equal to
frequency between the first- and second-order terms,
. Due to the difference in
can also be separated
between the first and second-order contributions
where
6.4
is the complex conjugate of
.
The Finite Element Model
The first-order problem can be treated in exactly the same way as in Chapter 3,
using the FEM described. In comparison, the second-order potentials
and the coefficients
,
and
and
can also be processed in the same way as the
first-order problem. For these problems the second-order cylindrical damper is
applied on
where
150
is the tangential coordinate, at
and is equal to
.
However, no Sommerfield-type boundary condition can be used for the
forced-wave problem due to the inhomogeneous surface boundary condition and the
phase-locked component of the potential
. Clark et al. (1991) derived a specific
radiation condition for a cylinder in waves, using asymptotic approximations in the
far field. Finding such approximations becomes difficult when considering more
complex bodies such as an OWC device and when the effects of the first-order
radiation wave need to be taken into account in the free surface forcing. We also
wish the model to be general enough to be able to deal with complex systems and/or
bathymetry. Numerical damping on the free surface was regarded as more
appropriate as this mechanism could absorb the outgoing scattered wave energy. A
damping term was therefore introduced in the free surface kinematic and dynamic
boundary conditions (6.2) and (6.3), between
,
where μ is the damping coefficient chosen as Bai et al. (2001)
After derivation, the expression of the boundary condition (6.25) becomes
151
Figure 6.2: Comparison of the free surface amplitudes
,
and
resulting from the model with the results from Chau & Eatock Taylor (1992), around
the circumference of a cylinder.
,
m,
and
. is the
azimuth of the cylindrical coordinates in degrees.
The need for a finite computational domain also means that the first-order
forcing terms have to be truncated at a certain distance
needs to be placed at a distance far enough from
from the cylinder.
then
in order to efficiently absorb the
energy of the wave. Tests were performed and convergence was found for
and
where
is the second-order wavelength,
. Using this numerical damper, the 3D FEM model was applied to the
forced-wave potential
and the forced-wave excitation coefficient
could be
152
derived through the volume flux inside the chamber using the equations (6.49) and
(6.53).
For the second-order problems, ten elements per second-order wavelength
were used. Due to these constraints and the increase of the domain size induced by
the implementation of the numerical damping zone, the resulting number of
elements, the number of nodes and therefore the CPU time increased significantly
compared to the first-order problems. The number of nodes and elements of the mesh
varied from around 80000 nodes and 50000 elements in low
120000 elements for high
to 190000 nodes and
. The CPU time varied from approximately 50 minutes
for low frequencies to around 150 minutes for high frequencies and additional runs
went from around 20 minutes for low frequencies to around 60 minutes for high
frequencies.
Several validations of the model prior to the study were carried out. One of
them is presented in Figure 6.2 where the dimensionless free surface amplitudes
and
around the circumference of a cylinder resulting from the model are compared with
the results from the semi-analytical solution developed by Chau & Eatock Taylor
(1992). This case can be considered as a special case of the current problem when
,
0,
and
. As one can see, the first and second-order
terms resulting from the model agree closely to the solution from Chau & Eatock
Taylor (1992) with an overall error on
of less than 3%. This provided
153
assurance that the extension from a linear model to second-order nonlinear model
was correct.
6.5
Results and Discussion
6.5.1
Dimensionless Parameters
Prior to presenting and discussing the different results obtained from the
model, we shall introduce the non-dimensional form of the properties of interest.
The non-dimensional free surfaces are defined as
The non-dimensional volume fluxes are chosen as
and
The dimensionless pneumatic damping coefficient is selected as
where
is the density of the air. At 20°C at sea level this is approximately equal
to 1.2kg.m-3.
Finally, the dimensionless mean hydrodynamic power extracted is defined as
154
and
where Cg is the group velocity of the first-order incident wave. It is important to note
that even if
is non-dimensionalized with a term in
due to the second-order term, whereas
6.5.2
, it is still dependent on
is not.
Results and Discussions
The dimensionless characteristics of the device considered in this study were:
inner cylinder radius a/h = 0.2, outer radius b/h = 0.25 and the wetted depth of the
cylinder D/h = 0.25. Contrary to the previous chapters, not all the hydrodynamic
coefficients are, in this problem, solely dependent on the frequency
wave excitation coefficient
this study,
and the forced-
is also dependent on the pneumatic damping . In
was either chosen equal to zero, representing the zero-pressure case, or
the optimum damping coefficient
for the first-order problem was considered.
Figure 6.3 and 6.4 present the various volume flux amplitudes computed for
0 and
through a range of frequencies
shows that when no pressure is present inside the chamber,
. In Figure 6.3
tends to 1 when
tends to 0, it passes through a maximum at the natural resonance frequency of the
device,
, and tends to 0 as
previously in Chapter 3. As
on the amplitudes of
increases. This is similar to the results found
is real and positive, the pressure has a damping effect
throughout the frequency range.
155
The second-order volume flux amplitude
possesses a more complex
behaviour than its first-order counterpart. Firstly, it is evident that
infinity as
diverges to
tends to 0. Such a result was anticipated and is a symptom of the
limitations of the second-order theory. The amplitude of the second-order wave
needs to stay smaller than the first-order amplitude, though the term
the second-order incident wave potential
for infinitesimally small
, in
, tends to infinity for low
and only
≤
can the assumption be fulfilled. In the area where
0.5, cnoidal wave theory would certainly be more appropriate. Results within this
frequency range will not be taken into account in the following discussion.
Figure 6.3: Amplitude of the dimensionless volume fluxes
different pneumatic damping coefficients
0, (
0) and
and
versus
for
.
156
a)
b)
Figure 6.4: Amplitude of the dimensionless volume fluxes versus
, and
.
for
0 and b)
,
,
, and
. a)
,
for
157
As
increases,
passes through two maxima before tending to 0 at high
frequencies. It can be shown that the first peak is situated at exactly half the
resonance frequency of the first-order volume flux,
peak is clearly situated at
. When
/2, whereas the larger second
,
is smaller than for
throughout the range of frequencies and the first peak, at
/2, disappears
completely.
In order to understand the behaviour of
, Figure 6.4.a and 6.4.b show the
amplitude of the different components,
, and
for
wave amplitude,
for
and
,
, respectively. Due to its dependency in the incident
diverges for low
/2 before tending to 0 for high
that
and
.
exhibits only one peak at
. In similarity with the first order, it is evident
/2 is the natural resonance frequency of the device for a wave oscillating at a
frequency 2 . As for the first-order situation, the diffracted wave amplitude
decreases as
increases away from resonance. Another reason for this decline
originates from the substantial decrease in the second-order incident wave amplitude.
For example, at
3, the component
The forced-wave amplitude
incident wave, does not diverge for low
order forcing is present,
resonance frequency
0.009.
, being independent of the second-order
but tends to 0. Even if no special first-
exhibits a maximum at the second-order natural
/2. This result may be related to the scattered free wave
components included in the forced wave. Even for small forcing, large responses in
amplitude are induced simply by the proximity of the OWC second-order natural
resonance frequency
/2. A significant result is that, for both
and
is the main component of the total second-order volume flux
peak situated at the first-order natural resonance frequency
,
around the
. This second peak is
evidently induced by the increase of the first-order forcings around
. Although
is not dependent on the second-order pressure, it displays lower values when
158
. It shows that the first-order forcings change significantly when the firstorder radiated wave is present.
The volume flux
low
related to the second-order radiated wave, diverges for
, and clearly has a destructive effect around the first peak at
. It can be
shown that this first maximum does not always disappear depending on the choice of
the pneumatic damping
Interestingly,
. As the frequency increases,
displays small values around
tends to zero.
, even if the total second-
order volume flux induces non-negligible second-order pressures. The values of the
second-order radiation conductance
and radiation susceptance
are very
small at this frequency and the response from the pressure is therefore also small.
One important outcome of this discussion is that when second-order theory is
applied to a resonant system such as an OWC, neither the forced nor the free waves
can be considered negligible. Such results can have significant implications in the
engineering domain. Second-order effects are often approximated through the use of
a first-order wave of twice the frequency with the same amplitude as the secondorder incident wave. Such an approximation would completely overlook the effect of
the forced wave.
Figure 6.5 presents the dimensionless free surface amplitudes
and
around and inside the chamber of the OWC device. The pneumatic damping
coefficient is
At the frequency
0 in Figure 6.5.a and 6.5.b, and
in Figure 6.5.c and 6.5.d.
3.0, it was found that the main component of the second-order
potential was the forced-wave element.
is seen to exhibit completely different
and more complex behaviour than its first-order counterpart for both
0 and
. One might have expected the amplitude of the second-order forced wave
to be especially high in areas of strong first-order amplitude. Apart from within the
chamber this is not the case and significant
can be found in areas of especially
low first-order amplitudes. High values of
are visible inside and close to the
chamber of the OWC device, but they are very localised and are significantly smaller
away from the device. Another noteworthy difference from the first-order situation is
159
presence of important large gradients in amplitude close to the outside of the OWC
and within OWC itself.
a)
b)
c)
d)
Figure 6.5: Dimensionless free surface amplitudes around and inside the OWC
device. a)
, b)
considering the pneumatic damping coefficient
0,
(
0) and c)
, d)
using the pneumatic damping coefficient
.
3.
To study the impact of the second order analysis on the mean power extraction,
was computed for different incident wave amplitudes and compared with the
first-order contribution
. The pneumatic damping was here chosen as
The results are presented in Figure 6.6. For small wave amplitudes,
is barely differentiable from
theory.
the system
.
0.01,
which is in agreement with the first-order
has a maximum value of around 0.8 located at the natural frequency of
.
tends to 0 as kh decreases or increases. However, as
160
increases, the contribution of the second-order mean power extraction becomes
significant. The most important effect of the second order is found for the maximum
value of
frequency
tested,
0.1, and it is situated slightly below the natural
. The increase in the maximum power extraction is around 20%. This
result is quite remarkable as it clearly shows that when an OWC is placed near to
shore, with relative high incident wave amplitudes, the effect of second-order terms
should not be overlooked and future studies on near-shore OWCs will have to
consider the non-linear effects of the system. It is especially interesting to note that
these significant second-order effects appear in a frequency zone where the incident
wave does not contain a significant second-order component.
Figure 6.6: Dimensionless mean hydrodynamic power extracted
and
various incident wave amplitudes
0.01, 0.02, 0.05 and 0.1 versus
pneumatic damping coefficient is
.
No peak in
appears at the frequency
for
. The
/2. This is not always the case and
in a preliminary study, Nader et al. (2012a), the second peak appeared when a
different pneumatic damping coefficient was chosen. This shows that
could
161
potentially be changed depending on the incident wave amplitude so as to adjust to
the second-order power extraction.
Optimisation of the OWC pneumatic damping was not performed in this study.
The main reason was that an energetic unbalance appears when applying the secondorder Stokes’ wave to an OWC device. Taking a frequency near the first-order
natural resonance frequency, it can be seen that the second-order incident wave
amplitude is particularly small and that its contribution in second-order energy is
negligible. It becomes evident that it is through the first-order forcing terms that the
important second-order effects appear.
We could then consider that these second-order effects are induced by a
transfer of energy from the first order. If first-order energy is transferred, then the
first-order energy should decrease, leading to a possible decrease of the first-order
power extraction ratio. Such behaviour was indeed observed in one recently
submitted study involving the present author (Luo et al. (2012)), for a twodimensional fully non-linear 2D model based on the Euler Equation using ANSYS
Fluent software. However, contrary to fully non-linear theory, Stokes’ wave theory
seems to fail in conserving the overall energy balance of the system by overlooking
the transfer of energy between the different orders.
In previous studies based on second-order theory, transfers between orders
have been considered negligible as long as the second-order forced wave stayed
small compared to the other components. However, in the present study, due to the
highly resonant characteristic of the OWC device, it was found that this assumption
was not valid. This represents a significant issue since quantification of energy flows
is the essential objective in the study of wave energy convertors.
6.6
Conclusion
This chapter has presented results on the contribution of the second-order non-
linear terms to the behaviour of an OWC system in relative small water depths and
with larger incident wave amplitudes. The relevant equations were derived using the
method of perturbation expansion applied to regular waves. The hydrodynamic
analysis was also extended to the second-order terms by deriving and evaluating the
introduction of second-order hydrodynamic coefficients.
162
After validation, the FEM model was then applied to a cylindrical fixed OWC
device with finite wall thickness. By deriving the different volume fluxes, it is shown
that within the second-order theory, the free-wave and especially the forced-wave
influences cannot be considered negligible.
The study was then directed to the contribution of the second-order terms in the
power extraction of the system as the incident wave amplitude increases. An
important finding is that this contribution can potentially be responsible for more
than 20% of the overall mean hydrodynamic power extraction. Such a result
demonstrates that when an OWC is placed near-shore, with relative high incident
wave amplitudes, the effect of second-order terms can become significant and that
future studies should certainly consider these non-linear effects for near-shore
OWCs.
However, the second-order Stokes’ wave theory an energy imbalance in the
direct application of was brought to light. It was shown that non-negligible first-order
energy could be transferred to the second order due to the importance of the forced
wave. Such effects would be especially strong around the first-order natural
resonance frequency of the system but they cannot be modelled using second-order
theory. As a consequence, these energy transfers could induce a significant decrease
in the first-order power extraction ratio. In the future, a method will need to be
developed in order to take into account these energy transfers.
163
7 POWER EXTRACTION OF A HEAVING OSCILLATING WATER
COLUMN DEVICE UNDER WEAKLY NONLINEAR WAVES
7.1
Introduction
It is interesting that contrary to the field of second-order wave forces acting on
fixed structures, only relatively few works have been published concerning secondorder forces on and motions of floating bodies in regular waves. Indirect methods
were developed by Pinkster & Van Oortmerssen (1977), Molin & Hairault (1983),
Molin & Marion (1986) and Matsui et al. (1992) whereas Moubayed & Williams
(1994) developed semi-analytical solutions for a floating cylinder body using an
Eigen function expansion approach. As stated by Moubayed & Williams (1994),
their work was mostly focused on providing an independent benchmark for the
testing of numerical methods.
This lack of interest originates from the fact that, up until now, floating bodies
such as boats or platforms have been specifically designed to have their motion
resonance frequency away from the natural wave spectra. It follows that the secondorder components were usually negligible compare to the first-order ones. Previous
studies focused mostly on the interactions between two or more waves where the
non-linear sum or difference of frequencies could induce resonance between
platform legs and/or non-negligible drift forces on one or multiple bodies (e.g.
Pinkster (1980), Kim & Yue (1990) and Kim (1998)). More recently, second-order
three dimensional models were developed in the time domain in order to more
accurately study these effects. These models are usually based on the BEM, as in
Cheung et al. (1993), or the FEM, as in Hong & Nam (2011).
On the other hand, for wave energy converters it is actually desired to have the
motion resonances within the incident wave spectra so as to extract maximum
hydrodynamic power. High motion amplitudes can directly induce non-negligible
non-linear effects.
In the previous chapter, the present author has found that second-order terms
become quite significant around the natural resonance frequency of the water column
of a fixed OWC device. A heaving OWC device presents further complexities as
compared to the fixed case. It possesses two motions interacting with each other and
each of these motions possesses distinct resonance frequencies, one for the water
column and one for the body, as seen in Chapter 5.
164
In this final analysis, second-order Stokes’ wave theory is extended to a freely
heaving OWC device, with a dynamic chamber pressure and the importance of the
second-order terms in the performance of the system is investigated.
7.1.1
Heaving Motion
Figure 7.1: Schematic diagram of the OWC device.
We consider the same system as in the previous chapter (cf. Figure 7.1).
However, the cylinder is, here, allowed to heave, i.e. has one degree of freedom. By
considering the displacement
of the device from the equilibrium position in the
direction, the kinetic boundary condition on the surface
at
, becomes
Following the method of perturbation expansion, we apply a Taylor series to
the boundary condition (7.1) in order to obtain the condition at the equilibrium
position (
)
165
The results of the Taylor series are made traceable by considering a
perturbation series on both
, as previously, and
with respect to the wave-
stiffness ,
giving
and
The solution for the displacement is also considered to be a set of a timeharmonics in regard to the incident wave frequency
added to a non-periodic
component and the complex values of the first-order
and second-order
of
the displacement are introduced
where the non-periodic component
is used to eliminate the constant in the
kinetic boundary (7.5). The kinetic boundary conditions (7.4) and (7.5) therefore
become
and
166
7.1.2
Boundary Conditions
The general boundary conditions for this problem can be expressed as follows:

In the outer domain
:
On the sea floor at
On the surface, at z = 0
and
The radiation condition on
when
tends to infinity
167
and a proper radiation condition for

Between the two regions at

In the inner region
on
:
:
On the sea floor at
On the device walls at
and
On the surface
and
On the surface, at
and
with
.
168
and
On the surface inside the chamber, at
and
and
7.1.3
Equation of Motion and Turbine Characteristics
Ignoring viscosity and moorings effects, the displacement of the device is
considered to follow the equation of motion
where
is the total mass of the device.
exerted by the hydrodynamic pressure on
pressure
corresponds to the hydrodynamic force
.
is the forced induced by the chamber
on the device at the top of the air chamber and is approximated to
169
At the hydrostatic equilibrium, the following relationship is obtained
By introducing the complex values of the velocity potentials, the displacements
and the chamber pressures, the equation of motion for the different orders can be
expressed as
and
The expressions of the dynamic forces
and
integrating the dynamic pressure components on the surface
of the device, the dynamic pressure on
can be found by
. Due to the heaving
is obtained by applying a Taylor series to
the Bernoulli equation followed by the perturbation expansion. One can then find the
expressions of the dynamic forces
and
as
and
As in the previous chapters, the drop in pressure through the turbine is
considered to be proportional to the volume flux through the turbine
170
Moreover, the pneumatic damping
is also considered to be real and positive
in this chapter.
By separating
into its first- and second-order complex terms
we obtain
However, the total volume flux going through the turbine is, here, dependent
on the heave motion of the device
where
is the volume flux created by the oscillation of the free surface inside the
OWC chamber,
and
is the volume flux created by the heaving motion of the device,
By introducing the complex values of the different variables, the relationship
expressions (7.32) become
171
and
The expressions of the volume fluxes
and
, induced by the oscillation
of the free surface inside the chamber, can be derived by expressing the free surface
components through the kinetic relation at the surface, giving
and
From the different relationships (7.26), (7.27), (7.36) and (7.37), the different
components of the displacement,
pressure,
fluxes,
and
and
and
, and the components of the chamber
, can be expressed in function of the water induced volume
, and the hydrodynamic forces,
and
,
172
and
The expressions for displacement and chamber pressure components can then
be introduced into the boundary conditions (7.17), (7.18), (7.21) and (7.22). For a
given geometry, incident wave frequency
, and pneumatic damping , the system
can now theoretically be solved, considering that a valid radiation condition was
applied for the second-order problem.
7.2
Analysis
The first-order potential can be analysed in the same way as in linear water
wave theory which was presented in Chapter 5. The fact that the pneumatic damping
coefficient
is, here, considered positive can simply be seen as the special case of
Chapter 5 where
and
displacement can then be expressed as
and
0. The first-order pressure and the first-order
173
where
,
Chapter 5.
,
,
,
,
,
,
and
,
are the same coefficients as defined in
and
geometry, on the incident wave frequency
are only dependent, for a given
.
The perturbation method, as seen in Section 7.2.2, has for effect to linearise the
second-order equations in term of second-order components (the velocity potential
, the pressure
and the displacement
) where the quadratic components
comprise first-order terms only. It is therefore possible to separate the velocity
potential
as follow
The diffracted wave potential
represents the effects of the interaction
between the incident wave potential
and the OWC device when the device is
fixed at its equilibrium position. After separation into the two numerical domains,
satisfies the general boundary conditions presented in Section 7.2.2 except that
the free surface boundary conditions (7.11), (7.20) and (7.22) and the kinematic
boundary condition (7.18) are replaced by homogeneous boundary conditions (the
quadratic first-order terms, the oscillating chamber pressure
displacement
and the
are all set to zeros).
The pressure radiated wave potential
is identical to the pressure radiated
wave potential of the linear wave problem and represents the effects, on the water, of
the oscillating chamber pressure
.
satisfies the general boundary conditions
presented in Section 7.2.2 by considering the potential
and
and the displacements
to be zero. The free surface boundary conditions (7.11), (7.20) and the
kinetic boundary condition (7.18) are also replaced by homogeneous boundary
conditions and the quadratic first-order terms in the free surface boundary condition
(7.22), inside the OWC chamber, are removed, leaving the pressure term only.
The heave motion radiated wave potential
is similar to the heave motion
radiated wave potential of the linear wave problem (cf. Chapter 4) and represents the
174
effects, on the water, of the motion of the OWC device.
satisfies the general
boundary conditions presented in Section 7.2.2 by considering the second-order
incident wave potential
and the oscillating chamber pressure
to be zero. The
free surface boundary conditions (7.11), (7.20) and (7.22) are also replaced by
homogeneous boundary conditions and the quadratic first-order term in the kinetic
boundary condition (7.18), on the surface
of the device, are removed leaving the
second-order displacement term only.
Finally, the forced-wave
relates to the effects induced by the forcing of
the quadratic first-order terms in the various boundary conditions.
satisfies the
general boundary conditions presented in Section 7.2.2 by considering the potential
, the pressure
and the displacement
to be zero.
differs from the
previous chapter (Chapter 6) due to the presence of the quadratic first-order term in
the inhomogeneous kinetic boundary condition (7.18), on the surface
of the
device.
The waves associated with the potentials
,
and
can propagate
freely on the free surface and satisfy the dispersion relation
where
is the wave number attached to free waves oscillating at frequency 2 .
However, for the same reasons as explained in Chapter 6, the forced-wave
associated with the velocity potential
The volume flux
where
does not satisfy (7.47).
inside the chamber can be separated into
175
and
The hydrodynamic force
can be separated into
where
and
By analogy with the linear-water wave theory, we can express the different
contribution of the diffracted wave potential
potential
, the pressure radiated wave
and heave motion radiated wave potential
where the coefficients
Similarly to the first-order,
,
,
,
,
,
,
,
and
,
and
as follow
are complex numbers.
are only dependent,
176
for a given geometry, on the incident wave frequency
.
,
would be
dependent on the direction of the incident wave propagation for a nonaxisymmetrical device, if considered. Moreover, due to the total similitude between
the second-order and first-order radiation problems, we can deduce the following
relationships
and
As a result of the quadratic first-order terms, the forced-wave contribution can
also be considered, in this problem, proportional to the square of the incident wave
amplitude
where
and
are the forced-wave volume flux response function and the
forced-wave hydrodynamic force response function, respectively.
,
are
dependent on the frequency , and on the pneumatic coefficient .
The second-order dynamic pressure and second-order displacement can then be
expressed as follows
177
and
The mean hydrodynamic power extracted by the system is defined by
being the first-order wave period equal to
. Due to the difference in
frequency between the first- and second-order terms,
can also be separated
between the first and second-order contribution
where
7.3
is the complex conjugate of
.
The Finite Element Model
The first-order problem can be treated in exactly the same way as in Chapter 4,
using the FEM described. The second-order potentials
,
and
can also
be processed in the same way as the first-order problem using the second-order
cylindrical damper at the outer limit of the domain applied to a free wave oscillating
at a frequency 2
and expressed in the relations (6.57) and (6.58).
The forced-wave potential
can processed similarly to forced-wave
potential from the previous chapter. The numerical damper described in equations
(6.62) in Chapter 6, was applied on the free surface, between
absorb the outgoing scattered wave energy.
, in order to
178
By use of the relations (7.49), (7.50), (7.52)-(7.54) and (7.57), the
hydrodynamic coefficients
,
,
,
,
,
and
can then be
derived by computing the hydrodynamic heave forces and the water volume flux
inside the chamber, for each of the different problems.
7.4
7.4.1
Results and Discussion
Dimensionless Parameters
Prior to presenting and discussing the different results obtained from the
model, we shall introduce the non-dimensional form of the properties of interest.
The non-dimensional free surfaces are defined as
The total non-dimensional volume fluxes are chosen as
and
The dimensionless pneumatic damping coefficient is selected as
And finally, the dimensionless mean hydrodynamic power extracted is
characterised by
179
and
where Cg is the group velocity of the first-order incident wave. Once again, it is
important to remind the reader that even if
, it is still dependent on
7.4.2
is non-dimensionalized with a term in
due to the second-order term whereas
is not.
Results and Discussions
In the following, the dimensionless characteristics of the device are the same as
in the previous chapter: inner cylinder radius a/h = 0.2, outer radius b/h = 0.25 and
the wetted depth of the cylinder D/h = 0.25. The hydrodynamic coefficients related to
the forced-wave
so
and
are dependent on the pneumatic damping coefficient
is first chosen equal to zeros, representing the case where no pressure is present
inside the chamber, then the optimum damping coefficient
for the first-order
problem is considered,
where
and
are the first-order overall radiation conductance and radiation
susceptance of the system satisfying
as previously seen in Chapter 5.
180
Figure 7.2: Amplitude of the dimensionless volume fluxes versus . a)
for different pneumatic damping coefficients
0, (
0) and
and
;
b)
for
,
, and
for
0; b)
,
,
, and
.
Figure 7.2 is shows the total volume flux amplitudes,
computed for
0 and
and
through a range of frequencies
,
. As
seen previously in Chapter 5, the first-order volume flux exhibits two maxima. The
first maximum, at
3.21, can be related to the resonance of the water column
whereas the second maximum, at
4.42, can be related to the resonance of the
heaving motion of the body. It can be seen that the pressure, using the first-order real
optimum damping coefficient,
, has for effect to damp the amplitude of
through the frequencies.
As expected, for
0 the second-order volume flux exhibits similar maxima
and band-width as the first-order amplitude at twice the first-order natural resonance
frequencies. Similarly to the previous chapter, these two peaks are almost fully
damped when
It is notable that, in this problem,
does not diverge at
181
low frequencies but tends to 0. The reason is that, at low frequencies, the body tends
to heave in phase with the water column. So, even if the amplitude of the secondorder incident wave diverges, the relative volume flux between the water column and
the heave motion decreases. That does not mean that the validity of the second-order
theory is extended to low frequencies.
a)
b)
c)
d)
Figure 7.3: Dimensionless free surface amplitudes around and inside the OWC. a)
, b)
considering the pneumatic damping coefficient
0, (
0N.m2
) and c)
, d)
using the pneumatic damping coefficient
.
3.
Around the first-order natural frequencies, we can observe non-negligible
values of the second-order volume flux. It can be shown that this behaviour is, once
again, the consequence of the forced wave. It is noticeable that
displays two
maxima around the first-order natural frequencies. However, these peaks do not
completely follow the first-order curve. The band-widths of the peaks are narrower
182
and
is small between the two first-order resonance frequencies. At the second
peak,
is smaller when a pressure is present in the chamber than when it is not.
However, at the first-order resonance of the water column,
becomes higher than
related to
related to
0, which is a particularly interesting
result. It could be shown that both the volume fluxes induced by the water column
and the motion of the device possess slightly lower values for
but the
relative motion is changed and its amplitude increases compare to the case where no
pressure is present inside the chamber.
From Figure 7.3, we can observe than the second-order amplitudes of the free
surface, for
3, is still localised within close proximity to the device. However,
in contrast to the fixed case, high values of
are still present when non-zero
pressure is considered.
Figure 7.4: Dimensionless mean hydrodynamic power extracted
and
various incident wave amplitude
0.01, 0.02, 0.05 and 0.1 versus .
for
As in the previous Chapter, the contribution of second order effects to mean
power extraction, is investigated by computing
for different incident wave
183
amplitudes. The results are also compared with the first-order contribution
pneumatic damping was chosen such that
Figure 7.4. For small amplitude waves,
differentiate from
. The
and the results are presented in
0.01,
is, as previously, difficult to
. But as the amplitude of the wave increases, the contribution
of the second-order power extraction becomes especially significant and even turns
out to be the main component of the total hydrodynamic power extraction, three
times higher than that of the first-order. This result is found for the maximum wave
amplitude tested,
0.1 at both natural resonant frequencies. These results show
that the second-order effects are, once again, especially important for resonant
systems. They are even more important for the floating OWC system because of its
multiple resonance frequencies and the coupling between the motion of OWC freesurface and structure. As discussed in Chapter 6, energy transfers from the first order
to second order are likely to be significant and could strongly impact on the firstorder power extraction ratio. Future studies will certainly need to take into account
both second-order motions and the associated energy transfers.
7.5
Conclusion
The present study is an extension of the investigation of second-order nonlinear
effects in the hydrodynamic and energetic behaviour from fixed to heaving OWC
devices. The perturbation expansion method was applied to the heaving motion in
order to derive the boundary conditions at the mean-position of the body. Secondorder hydrodynamic coefficients were also introduced and discussed.
The second-order 3D FEM model was then applied to a cylindrical freelyheaving OWC device with finite wall thickness and the coefficients computed from
the model results.
As for the fixed case, the second-order forced wave was found to induce
important volume fluxes around the first-order natural resonance frequencies of the
system. Interestingly and contrary to the fixed case, the pressure was observed to
have a positive impact in the second-order relative volume flux around the first-order
resonance frequency of the water column. Even more remarkable, the second-order
contributions in the hydrodynamic power extraction could become the main
184
component of the total power extracted by the system. Values as high as three times
that of first-order were predicted when large wave amplitudes were considered.
It follows that the second-order contributions can be particularly important in
this configuration and future studies will certainly have to investigate these nonlinear effects when studying performance of floating OWC devices.
185
8 CONCLUSIONS AND RECOMMENDATIONS
During the course of the present study, a new FEM model has been
successfully developed to study the hydrodynamic and energetic behaviour of OWCs
in ocean waves, and applied to the analysis of single isolated devices and arrays of
such devices.
By applying and extending mathematical hydrodynamic theories to a set of
new situations, the current work has brought significantly greater understanding of
the hydrodynamic and energetic performance of such complex systems.
Chapter 2 focused on the effect of wave interactions on power-capture
efficiency of finite arrays of OWC devices. The study showed that such interactions
may be roughly classified into three categories depending on the relative magnitude
of the wavelengths to the structure non-dimensionalised array spacing. When the
wavelength is large relative to the array spacing, then the array behaviour most
closely resembles that of a single large device. At the other extreme, when the
wavelength is small relative to the array spacing, the behaviour of the devices in the
array tends toward that predicted for a Single Isolated OWC Device (SIOD). In
between these extremes, the interaction effects are strong and the performance of the
array and its constituent devices are highly dependent on the spacing and location
within the configuration.
The results indicated that optimised placement of the OWCs in an array can
improve the mean power capture efficiency whereas other configuration might have
a destructive effect in the power absorption. The trials conducted at various wave
incidence angles in the present study suggest that, for the configurations of array
studied, the array should be placed in alignment with the dominant wave direction
for maximum array efficiency. The capture width efficiency was found to be
generally higher for small array spacing configurations. This has important practical
consequences in the design of multiple-chamber devices. The optimal pneumatic
damping for individual OWC chambers in an array was also shown to deviate from
that predicted for a SIOD and could also vary for devices within a given array.
Chapter 3 considered a single fixed OWC device. Air compressibility inside
the chamber was taken into consideration and a more in-depth analysis of the
hydrodynamic characteristics was presented. It was shown that only a few specific
186
hydrodynamic coefficients, for a given geometry and incident wave frequency, are
needed in order to determine the dynamic and energetic behaviour of an OWC device
in waves. Pressures, volume fluxes, power extractions and power capture widths
could then be directly obtained for any desired parameter of the turbine and chamber
volume of air. Optimum turbine parameter, leading to maximum hydrodynamic
power extraction and maximum capture width, could also be directly computed for
any given chamber volume of air.
The study especially focused on the consequences of changing the device’s
geometrical properties. It was observed that due to air compressibility, the chamber
volume of air had significant effects on the dynamic and energetic performance of
the device. Influences of the change in draft, inner radius and wall thickness were
also studied. It was found that each of these parameters have a large effect on the
position of the natural frequency of the system, on the amount of hydrodynamic
power extraction and on the frequency band-width.
Chapter 4 was the natural extension of the preceding two chapters where the
method developed in Chapter 3 was extended to the study of the dynamic and
energetic performance of finite arrays of fixed OWC devices. The different
influences between devices were explicitly described and air compressibility was
also considered.
Following this method, the FEM model was applied to the study of three
different array configurations: a column of two identical cylindrical OWC devices, a
row of two identical cylindrical OWC devices, and two rows and two columns of
identical cylindrical OWC devices. As in Chapter 2, an array of devices was found to
behave quite differently to a single isolated OWC device. It was then demonstrated
that the inner properties of the OWC devices and the radiation influences between
devices are strongly dependent on the position of the device in the array. It was also
revealed that the dynamic and energetic performance of the system becomes more
complex and distinct from the SIOD as the number of devices in the array increases.
As in Chapter 2, more hydrodynamic power could be extracted from the array,
at some frequencies, than it would be extracted from the sum of the same number of
isolated devices. Optimisation of the overall power extraction of the system was
performed. A closed form of the optimum parameters was derived in the first
187
problem studied. However, due to the significant increase in complexity of the
overall mean hydrodynamic power expression, an optimisation was then performed,
in the two other problems, through the use of the Nelder-Mead Simplex Method
applied to the total mean capture width of the system. It was shown that taking into
consideration the coupling between devices increases the overall power extraction of
the system. The results also suggest that the position of the device in the array should
be taken into account when determining device parameters so as to increase the
maximum power extraction of the system or the overall frequency power-capture
band-width.
OWC devices can also be floating structures. The motion of the structure
becomes an important factor in the power extraction of the system. Chapter 5 looked
at a single OWC device that is allowed to heave, i.e. the motion has one degree of
freedom. The method of interaction between oscillating systems was applied
specifically to the heaving OWC device and extended to take into account the
relationship between the volume flux and the pressure. Direct coupling between the
motion of the device, the pressure inside the chamber, the volume fluxes and the
forces were considered. Moreover, air compressibility and the spring effect of the
moorings system were taken into account.
As previously, a study of the dynamic and energetic behaviour of the OWC
device was performed through a set of frequency dependent hydrodynamic
coefficients. Pressures, volume fluxes, heave motion and power extraction were then
directly computed for any desired parameter of the turbine, chamber volume of air,
and mooring properties. The closed form of the optimum parameter of the turbine,
for maximum energy extraction, could also be derived.
The FEM model was then applied to a cylindrical OWC device. In the first part
of the study, the device was considered to be freely floating and the effect of air
compressibility was investigated. It was found that air compressibility can increase
the power capture band width of the system. This band-width can be significantly
broader than for the fixed case but the values of the capture width were, on average,
considerably lower. The exceptions were the two narrow peaks observed at the
heaving motion and air compressibility resonance frequencies.
188
In the second part of the study, the influence of the mooring system on the
performance of the device was investigated by varying the restoring forced
coefficient. The most significant outcome was the appearance of a new maximum in
the power capture width. This maximum could considerably widen the power capture
width compared to the fixed case and could induce significantly higher power
extraction rate compared to a freely floating device. These results demonstrate the
non-negligible influences that the mooring system can have on the dynamic and
energetic behaviour of a floating OWC device. The mooring system therefore
becomes an important parameter of the overall system performance and can be
designed so as to improve power extraction.
From each of these chapters, it was shown that the hydrodynamic power
absorbed by the system is strongly dependent on a large number of parameters
depending on whether a single device is deployed or whether arrays of these devices
are considered and depending on whether the system is fixed or floating. Each of
these parameters (device geometry, chamber size, turbine characteristics, distance
between devices, mooring characteristics, etc.) can be specified to best describe the
energetic behaviour of the particular system in connection with a local wave climate.
The various studies herein have demonstrated the potential of the new FEM model to
efficiently and accurately assist such analysis.
In Chapters 6 and 7, the hydrodynamic analysis and the FEM model were
extended using Stokes’ wave theory up to second order. The relevant equations were
derived using the perturbation expansion method applied to regular waves and
second-order hydrodynamic coefficients were introduced and evaluated.
The second-order 3D FEM model was then applied to a cylindrical fixed OWC
device with finite wall thickness. By deriving the different volume fluxes, it was
shown that within the second-order theory, the free-wave and especially the forcedwave influences cannot be considered negligible.
The study was then directed to the contribution of the second-order terms in the
power extraction of the system as the incident wave amplitude increases. An
important finding was that this contribution can potentially be responsible for more
than 20% of the overall mean hydrodynamic power extraction.
189
In Chapter 7, the study extended the investigation of second-order nonlinear
effects in the hydrodynamic and energetic behaviour from a fixed to a heaving OWC
device. The second-order 3D FEM model was applied to a cylindrical device with
finite wall thickness and the coefficients computed from the model results.
As for the fixed case, the second-order forced wave was found to induce nonnegligible volume fluxes around the first-order natural resonance frequencies of the
device. Interestingly and in contrast with the fixed case, the pressure inside the
chamber could increase the second-order relative volume flux around the first-order
resonance frequency of the water column. Even more remarkable, the second-order
contributions in the hydrodynamic power extraction could become the main
component of the total power extracted by the system. Values as high as three times
that of first-order were predicted when large wave amplitudes were considered.
Both Chapter 6 and 7 demonstrated that the second-order contributions could
become particularly important when studying performance of fixed and floating
OWC devices and that future studies related to this type of systems should certainly
consider these non-linear effects.
However, an energetic imbalance in the direct application of the second-order
Stokes’ wave theory was brought to light. It was shown that non-negligible firstorder energy could be transferred to the second order due to the importance of the
forced wave. Such effects would be especially strong around the natural resonance
frequencies of the system although they cannot be captured using Stokes’ theory. As
a consequence, these energy transfers could lead to a significant error in the power
extraction ratio predicted if the first-order theory is used.
For all the various analyses presented in this thesis, the OWC devices were
chosen to be hollow circular cylinders of finite wall-thicknesses. This shape was
chosen as being the most simple in order for the research to be the most general
possible, while maintaining a close relationship with the physical constraints of
practical systems. The model was developed to not only deal with this fundamental
geometry, but also to analyse more complex shapes and/or possible bathymetries.
Moreover, the research focused especially on the device behaviour and since the
model was a direct method, a wide range of information including flow rates,
velocities and pressures throughout the fluid, could be determined. The model can
190
therefore be applied to provide other types of information such as determination of
the forces on the devices and/or the impact on the environment surrounding the
installation of such systems. Specific studies of more complex systems were in fact
performed for the project partner Oceanlinx Ltd. However, by reason of their
confidential character, they have not been included in this manuscript.
Further research in the area related to OWC devices is undoubtedly needed.
Following the work performed during the PhD position and the results presented in
the thesis, several areas of research could be improved in the future.
The first step would certainly be to compare the model result with experimental
work. The turbine performance curved and continuous wave spectra should also be
included in the analysis in order to obtain a more practical optimisation of the
parameters.
Concerning floating devices, some systems were developed with an underwater opening at the front like the Oceanlinx Ltd MK3 pre-commercial system or at
the rear such as the OceanEnergy Ltd OE Buoy. Effects of the pitch and surge
motions can become significant in the performance of the system and should
certainly be investigated.
In the second-order domain, an important point would be to derivate a more
accurate method in order to better quantify the energy transfers between orders. In
this way, power losses induce by non-linear effect could be measured and possible
corrections on the optimum turbine properties could be derived depending on the
incident wave amplitude.
Moreover, the model was developed in the frequency domain which is
sufficient when considering linear or second-order water wave theory and linear
power-take-off. However, different types of turbine, such as the Denniss-Auld
turbine used by Oceanlinx Ltd (c.f. Curran et al. (2000), Finnigan & Alcorn (2003),
Finnigan & Auld (2003)), can induce more complex relationship between the volume
flux and the pressure inside the chamber of the OWC device. In order to more
accurately model the influence of the turbine, a time-stepping numerical model could
be developed with the implementation of a more precise pressure/volume-flux cycle.
Furthermore, when the second-order Stokes’ wave theory is applied in the frequency
domain, the non-linear interactions between different wave frequencies are
191
overlooked and a time-stepping model could also be of interest in order to investigate
the effect of these interactions.
Finally, in order to the study the performance of the system in more extreme
environments or to investigate the effect of turbulence, fully 3D non-linear water
wave models should be applied.
These types of model already exist and were applied to body-wave studies as in
Causon et al. (2008) and Agamloh et al. (2008), but they require a large amount of
memory and CPU time limiting the amount of studies they can perform.
192
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